Friday, February 3, 2017

Hashing and Probability of Collision

Problem D of Codeforces #395 required hashes of rooted trees. In this post I give some examples of good hashes and bad hashes, and then construct a good hash for rooted trees.

The most well-known hash in competitive programming is probably the rolling-hash of strings. The following problem looks stupid, but let's solve it using rolling hash anyway.

Problem 1. You are given two strings s, t of lengths N (<= 10^5) that consist of lowercase letters. Determine if they are the same.

Let MOD=10^9+7. Choose an integer r uniformly at random from the interval [0, MOD). Compute the sum of r^i * s[i] over all i. This is called the rolling hash of s. We can consider it as a polynomial of r, let's denote it as S(r). Similarly, define T(r). The hash collision happens when s and t are different strings (thus S and T are different polynomials), but unluckily S(r) = T(r). Here we use Schwartz-Zippel lemma:

Let P(x_1, ..., x_k) be a non-zero (multi-variable) polynomial on a finite field F_MOD. If the variables are chosen independently and uniformly at random, the probability that P(x_1, ..., x_k) = 0 is at most D/MOD, where D is the total degree of the polynomial.

In the case of rolling hash, we are only interested in single-variable polynomials. From the lemma above, we can prove that the collision probability is at most N/MOD (<= 1/10^4). Thus I'd say this is a "good hash". My intuition tells that in practice the probability is as small as 1/MOD, but that looks at least very hard to prove. Let me give an interesting example that shows the intuition is not always correct. In the lemma F_MOD must be a field, and MOD must be a prime. What happens if the modulo is not prime, for example 2^64 (this would simplify the implementation). It turns out that the rolling hash doesn't work well for a certain case if the modulo is 2^64. Detailed description on Codeforces.

Problem 2. You are given two graphs G, H with N (<= 10^3) vertices and M (<= 10^4) edges. Determine if they are isomorphic.

For each vertex v in the graph and an integer d, define the level-d hash for v, h(d, v). When d = 0, define h(0, v) = 1. Otherwise, let x_1, ..., x_k be the sorted list of level-(d-1) hashes of vertices adjacent to v, and define h(d, v) as the sum of x_i * r^i modulo MOD. Compute the level-N hashes of the two graphs for each vertex, sort them, and compare them.

It looks very hard to find a testcase against this hash. (Actually no - see Petr's comment below). However there's no theoretical guarantee under this hash and I think most people agree that this doesn't look valid as an intended solution of a problem in a contest (graph isomorphism is a well-known difficult problem). I'd say this is a "bad hash".

OK, now we want to construct a good hash for rooted trees. Before that, let's try an easier version. Hashes for (multi)sets.

Problem 3. You are given two (multi)sets A = {a_1, ...., a_n} and B = {b_1, ..., b_n}. n <= 10^5 and each element is in the interval [0, MOD). Determine if they are the same.

One possible solution is to sort the input and then compute the rolling hash of it. Can you find a linear-time hash? The solution is indeed very simple. Let's take a random integer r from the interval [0, MOD), and compute the hash (r+a_1)(r+a_2)...(r+a_n). This is a polynomial of r of degree n, so again from Schwartz-Zippel lemma, the collision probability is at most n/MOD.

Problem 4. You are given two rooted trees with n vertices. Determine if they are isomorphic.

Like previous examples, we assign a polynomial for a rooted tree, and evaluate the polynomial for random variables. However, unlike previous examples, this time we use multi-variable polynomials. We define the polynomial P(G) for a rooted tree G as follows. If G is a single leaf, P(G) = 1. Otherwise, let d be the depth of the tree, k be the number of children of the root of G, and G_1, G_2, ..., G_d be the subtrees corresponding to those children. Define P(G) = (x_d + P(G_1))(x_d + P(G_2))...(x_d + P(G_k)). See the picture for an example. (x_1, x_2, x_3 are replaced with x, y, z).

This way we get a multi-variable polynomial over d variables of degree l, where l is the number of leaves in the tree. We can prove that two polynomials that correspond to non-isomorphic trees are different (by using the uniqueness of factorization of the ring of polynomials). Thus, if we evaluate this polynomial for random variables, we get a good hash with collision probability at most l/MOD!


53 comments:

  1. Awesome explanation!

    For graph isomorphism, it's easy to break such hashing: take two non-isomorphic regular (all degrees the same) graphs.

    ReplyDelete
    Replies
    1. Thank you for pointing out, I'll fix it.
      Maybe computing hashes of graphs of this size is hard even in practice.

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  2. "It turns out that the rolling hash doesn't work well for a certain case if the modulo is 2^64. Comments in Russian." Comment link is broken Can you please fix it.
    Thanks

    ReplyDelete
    Replies
    1. I found an English description: http://codeforces.com/blog/entry/4898

      Delete
    2. It's not exactly broken, just change the language in codeforces to russian and click the link again :)

      Delete
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  3. "and G_1, G_2, ..., G_d be the subtrees corresponding to those children"

    I think it should be G_1, G_2, ..., G_k. Am I right?

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  14. I am, Susan Wanda from USA, North Carolina I promise to share this testimony all over the world once my Husband return back to me, and today with all due respect I want to thank dr. Odenore for bringing joy and happiness to my relationship and my family. I want to inform you all that there is a spell caster that is real and genuine. I never believed in any of these things until i loosed my husband, I required help until i found a great spell caster, And he cast a love spell for me, and he assured me that I will get my husband back in two days after the spell has been cast. Three days later, my phone rang, and so shockingly, it was my husband who has not called me for past 6 months now, and made an apology for the heart break, and told me that he is ready to be my back bone till the rest of his life with me. Dr Odenore, released him up to know how much I loved and wanted him. And opened his eyes to picture how much we have share together. As I`m writing this testimony right now I`m the most happiest girl on earth and me and my husband is living a happy life and our love is now stronger than how it were even before our break up. So that`s why I promised to share my testimony all over the universe. All thanks goes to Dr Odenore for the excessive work that he has done for me. Below is the email address in any situation you are undergoing a heart break, and I assure you that as he has done mine for me, he will definitely help you too. contact him now on email odenoretemple@gmail.com

    (1) Cancer spell
    (2) Pregnancy spell
    (3) Hiv/aids spell*
    (4) Getting your scam money back*
    (5) If you want a child.
    (6) if you always have bad dream
    (7) Spell to Get your love one out of jail .
    (8) Get a huge loan without paying any fee spell*
    (9) You want women/men to run after you.
    (10) If you want your ex back.
    Contact him today for your help and you will be happy for ever. REPLY (email : odenoretemple@gmail.com ) WhatsApp him +2349068901741

    ReplyDelete
  15. Online Solution To Love Problem ( He Is Expert Astrologer In This ) +91 8764361313
    Girlfriend Problem Solution
    Love Problem Solution

    ReplyDelete
  16. Online Solution To Love Problem ( He Is Expert Astrologer In This ) +91 8764361313
    Girlfriend Problem Solution
    Love Problem Solution
    Love Marriage Solution

    ReplyDelete
  17. Real And Urgent Spell Caster Reviews To Solve Your Relationship Problem Dr Davi..

    I am from Canada, I have great joy in me as i am writing this testimony about the great man called Dr. Davi When my lover left me i never taught that i will be able to get her back after all she put me through, But i am so happy that after the interference of Dr. Davi i was able to get my lover back within 48hours. Are you going through a break-up or divorce? Do you need your Ex back fast? Don't cry anymore, contact Dr Davi powerful Love spell that work fast now!! I was hurt and heart broken when my lover breakup with me, and i could not know what next to do again, I love my girlfriend so much but she was cheating on me with another man and this makes her break up with me so that she can be able to get married to the other man and this guy use witchcraft on my girlfriend to make her hate me and our kids and this was so critical and uncalled-for. I went to Florida to see a friend and who was having the same problem with me but she later got her husband back with the Same Dr. Davi love spell. She told me that Dr Davi help with love spell in getting back lost lover back, and i decided to contact the same Dr Davi and he told me what is needed to be done for me to have my woman back. To my surprise my lover really called me and told me that she miss me so much, Oh My God! i was so excited, and today i am happy with my woman again and we are joyfully living together as one big family and i thank the powerful spell caster Dr Davi for bringing back my lover and i decided to share my story on the internet that Dr Davi is the best spell caster online who i will always pray to live long to help his children in the time of trouble, If you have any problem contact Dr Davi and i guarantee you that he will help you,Email him at:

    Email: Spelltemple@fastmail.com
    Call him: +55 (11) 97732-4658   

    ReplyDelete
  18. Real And Urgent Spell Caster Reviews To Solve Your Relationship Problem Dr Davi..

    I am from Canada, I have great joy in me as i am writing this testimony about the great man called Dr. Davi When my lover left me i never taught that i will be able to get her back after all she put me through, But i am so happy that after the interference of Dr. Davi i was able to get my lover back within 48hours. Are you going through a break-up or divorce? Do you need your Ex back fast? Don't cry anymore, contact Dr Davi powerful Love spell that work fast now!! I was hurt and heart broken when my lover breakup with me, and i could not know what next to do again, I love my girlfriend so much but she was cheating on me with another man and this makes her break up with me so that she can be able to get married to the other man and this guy use witchcraft on my girlfriend to make her hate me and our kids and this was so critical and uncalled-for. I went to Florida to see a friend and who was having the same problem with me but she later got her husband back with the Same Dr. Davi love spell. She told me that Dr Davi help with love spell in getting back lost lover back, and i decided to contact the same Dr Davi and he told me what is needed to be done for me to have my woman back. To my surprise my lover really called me and told me that she miss me so much, Oh My God! i was so excited, and today i am happy with my woman again and we are joyfully living together as one big family and i thank the powerful spell caster Dr Davi for bringing back my lover and i decided to share my story on the internet that Dr Davi is the best spell caster online who i will always pray to live long to help his children in the time of trouble, If you have any problem contact Dr Davi and i guarantee you that he will help you,Email him at:

    Email: Spelltemple@fastmail.com
    Call him: +55 (11) 97732-4658   

    ReplyDelete
  19. i everyone here I have just experience the wonders of Dr.Odiase love spell, that have been spread on the Internet and worldwide, How he marvelously helped people all over the world to restored back their marriage life and get back lost lovers, and also help to win lottery. I contacted him after going through so many testimonies from different people how he help to bring back ex lover back, i told him about my husband that abandoned me about 7 years ago, and left home with all i had.. Dr.Odiase
    only told me to smile and have a rest of mind he will handle it all in just 24 hours, After the second day my husband called me he was begging I was just so shocked, i pick the call and couldn't believe my ears, he was really begging me to forgive him and making promises on phone.. He came back home and also got me a new car just for him to proof his love for me. i was so happy and called Dr.Odiase and thanked him, he only told me to share the good news all over the world. ( He can also treat other similar spells)
    Love Spells
    Lottery Spells
    winning court case
    Divorce spells
    Success
    Binding spells
    Money rituals
    Low sperm count
    Infertility in women
    Breast enlargement/reduction
    Penis enlargement/reduction
    Health, Well Being
    Protection, Healing
    Curses, Hex, Breakups
    NEW! Combo Spells
    High Priestess Spells
    Vampire Spells
    Authentic Voodoo Spells
    Custom, Other Spell,
    You can reach him via email:odiasetusha@gmail.com or whatsapp +2348069079544

    ReplyDelete
  20. Hello my name is Kallya from USA i want to tell the world about the great and mighty spell caster called Priest Ade my husband was cheating on me and no longer committed to me and our kids when i asked him what the problem was he told me he has fell out of love for me and wanted a divorce i was so heart broken i cried all day and night but he left home i was looking for something online when i saw an article how the great and powerful Priest Ade have helped so many in similar situation like mine he email address was there so i sent him an email telling him about my problem he told me he shall return back to me within 24hrs i did everything he asked me to do the nest day to my greatest surprise my husband came back home and was crying and begging for me to forgive and accept him back he can also help you contact
    ancientspiritspellcast@yahoo.com or ancientspiritspellcast@gmail.com

    Blogs: https://kallya1.blogspot.com

    Website http://ancientspiritspellcast.website2.me

    WhatsApp +2347059715465

    ReplyDelete
  21. Hello my name is Kallya from USA i want to tell the world about the great and mighty spell caster called Priest Ade my husband was cheating on me and no longer committed to me and our kids when i asked him what the problem was he told me he has fell out of love for me and wanted a divorce i was so heart broken i cried all day and night but he left home i was looking for something online when i saw an article how the great and powerful Priest Ade have helped so many in similar situation like mine he email address was there so i sent him an email telling him about my problem he told me he shall return back to me within 24hrs i did everything he asked me to do the nest day to my greatest surprise my husband came back home and was crying and begging for me to forgive and accept him back he can also help you contact
    ancientspiritspellcast@yahoo.com or ancientspiritspellcast@gmail.com

    Blogs: https://kallya1.blogspot.com

    Website http://ancientspiritspellcast.website2.me

    WhatsApp +2347059715465

    ReplyDelete
  22. BROKEN HEART MADE WHOLE AGAIN.....!!!!!!!!
    I never had a thing nor belief with spell casters until I saw a need for it when someone so dear to my heart left me for another man who doesn't even give a damn about her. Thanks to Dr. Odinani Owelle who used a spell to bring back my girlfriend to me in just two days when she came weeping and asking I forgive her for ever letting go and that she was gonna make up every bit of lost time and make me happier than ever for making me sob one bit
    I came in contact with Dr. Odinani Owelle in a blog and thought of giving it a trial because I could not imagine that someone I love with all my heart was gonna leave me right under my close watch, the and then did I get to know that my close watch was not good enough. Like promised, Dr Odinani Owelle kept to his just words that brought back my love home to daddy and I am gonna love her more.
    Wanna reach out to all the broken homes and relationships via this post of mine to believe that their problems can always be fixed, just get the right person to do it and I testify that there is no other right person as Dr. Odinani Owelle the real spell cast in the world Contact Him on hi's email adders odinanisolutioncentre@gmail.com.
    1. Getting your lover or husband back
    2. Spiritual bulletproof
    3. Training
    4. Money spell
    5. Long life spell
    6. Prosperity spell
    7. Protection spell
    8. Get a job spell
    9. Becoming a manager spell
    10. Get a huge loan without paying any fee spell
    11. Getting your scam money back
    12. Child spell
    13. Pregnancy spell
    14. Freedom spell
    15. Love spell
    16, vanishing spell
    17. Invisible human spell
    18. Success or pass spell
    19. Marriage spell
    20. Avenging spell
    21. Popularity spell
    22. Killing spell
    23. Cancer spell...........

    odinanisolutioncentre@gmail.com.

    ReplyDelete
  23. It just occur to me that I have not done the right thing since when my husband came back to me, I am on this blog to give thanks to whom it deserve, Some couples of weeks ago my life was in a terrible shape because my husband left me and I never believe that I was going to get him back, But through the help of this powerful spell caster called Dr.Ekpen my life is now in a joyful mood, I must recommend the services of Dr.Ekpen to anyone out there that they should contact Dr.Ekpen through these details below: ( ekpentemple@gmail.com ) or whatsapp +2347050270218 because through Dr.Ekpen assistance my marriage was restored.

    ReplyDelete
  24. I will recommend anyone looking for Business loan to Le_Meridian they helped me with Four Million USD loan to startup my Quilting business and it's was fast When obtaining a loan from them it was surprising at how easy they were to work with. They can finance up to the amount of $900,000,000.000 (Nine Hundred Million Dollars) in any region of the world as long as there 1.9% ROI can be guaranteed on the projects.The process was fast and secure. It was definitely a positive experience.Avoid scammers on here and contact Le_Meridian Funding Service On. lfdsloans@lemeridianfds.com if you looking for business loan,Thanks.
    Leslie,

    ReplyDelete
  25. Today has being the most happiest day of my life after 5 year of sadness and sorrow without being with the one i love, i tried all my possible best to make sure i make my lover happy but it never seems to work out well it was like am doing everything in vain but all thanks to Dr.Voodoo for coming to change all my worries and sadness to Joy. I knew the great man when i read some wonderful reviews about Him how he has helped a lots of people on there relationship problem i was reading a magazine which then i saw great testimonies as well which then i decided not to waste time. He told me not to worry that he assures me that within 3day everything would be sorted out i believed Dr.Voodoo so much because i believe he can't fail me and i sent him all my details. Truly Dr.Voodoo never failed me,my husband who left me for good a year come back to me. My husband and i have been living contentedly since this spell caster reunited us together with his love spells. Thank you so so so much Dr.Voodoo for your powerful spells. Expressions are not sufficient to say thank you, here is his email address(voodoospell10@gmail.com) or call (+2348135790702) for more information or WhatsApp with him.....

    ReplyDelete
  26. TIRED OF HEART BREAK, YOU WANT YOUR RELATIONSHIP TO BE RESTORED EMAIL HIM ON (Drjoseph818@gmail.com) or whatsapp him with +2347031663661

    Am Aeron Berry from Sweden and am here to give a testimony about Dr Joseph who actually reunion my marriage of 2 years with is incredible spell. It was a thing of joy to see my wife who divorced over 2years ago to come back begging for forgiveness. It was a difficult time for me as a father to actually take of my two daughters without their mother and the space where actually affecting our daughters cause the can't talk girls issues with me but their mom. A big thanks to Gladys Kim my daughters therapist who I actually who I employed to fill their mom space. Hoping that they can share their thoughts and challenges to her but it didn't work. She introduced me to Dr Joseph who always help her with a spell to increase her therapy skills which she do use for her clients when necessary. It only took Dr Joseph 40 hours to finish the spell for me and now am a testimonial.

    Email Him if you are into similar problem.
    DOCTOR JOSEPH CAN AS WELL HELP WITH THE FOLLOWING PROBLEMS
    1. HIV/AIDS SPELL
    2. HERPES SPELL
    3. CANCER SPELL
    4 PENIS ENLARGEMENT CREAM SPELL
    5 PREGNANCY SPELL

    ReplyDelete
  27. TIRED OF HEART BREAK, YOU WANT YOUR RELATIONSHIP TO BE RESTORED EMAIL HIM ON (Drjoseph818@gmail.com) or whatsapp him with +2347031663661

    Am Aeron Berry from Sweden and am here to give a testimony about Dr Joseph who actually reunion my marriage of 2 years with is incredible spell. It was a thing of joy to see my wife who divorced over 2years ago to come back begging for forgiveness. It was a difficult time for me as a father to actually take of my two daughters without their mother and the space where actually affecting our daughters cause the can't talk girls issues with me but their mom. A big thanks to Gladys Kim my daughters therapist who I actually who I employed to fill their mom space. Hoping that they can share their thoughts and challenges to her but it didn't work. She introduced me to Dr Joseph who always help her with a spell to increase her therapy skills which she do use for her clients when necessary. It only took Dr Joseph 40 hours to finish the spell for me and now am a testimonial.

    Email Him if you are into similar problem.
    DOCTOR JOSEPH CAN AS WELL HELP WITH THE FOLLOWING PROBLEMS
    1. HIV/AIDS SPELL
    2. HERPES SPELL
    3. CANCER SPELL
    4 PENIS ENLARGEMENT CREAM SPELL
    5 PREGNANCY SPELL














    TIRED OF HEART BREAK, YOU WANT YOUR RELATIONSHIP TO BE RESTORED EMAIL HIM ON (Drjoseph818@gmail.com) or whatsapp him with +2347031663661

    Am Aeron Berry from Sweden and am here to give a testimony about Dr Joseph who actually reunion my marriage of 2 years with is incredible spell. It was a thing of joy to see my wife who divorced over 2years ago to come back begging for forgiveness. It was a difficult time for me as a father to actually take of my two daughters without their mother and the space where actually affecting our daughters cause the can't talk girls issues with me but their mom. A big thanks to Gladys Kim my daughters therapist who I actually who I employed to fill their mom space. Hoping that they can share their thoughts and challenges to her but it didn't work. She introduced me to Dr Joseph who always help her with a spell to increase her therapy skills which she do use for her clients when necessary. It only took Dr Joseph 40 hours to finish the spell for me and now am a testimonial.

    Email Him if you are into similar problem.
    DOCTOR JOSEPH CAN AS WELL HELP WITH THE FOLLOWING PROBLEMS
    1. HIV/AIDS SPELL
    2. HERPES SPELL
    3. CANCER SPELL
    4 PENIS ENLARGEMENT CREAM SPELL
    5 PREGNANCY SPELL

    ReplyDelete
  28. What a wonderful and a straight forward spell caster that has brought back joy and happiness into my life!!! Am giving this testimony because am so happy, I want to thank Dr. dodo for the great thing He has done in my life , He brought happiness back to my life, I never believed in spell casters until my life fell apart when my lover of 6 years decided to call it quit almost when we wanted to get married. I was so emotional breakdown to the extent i could not do anything reasonable again, after 2months in pain before an old friend of mine introduced me to a spell caster on line called Dr dodo, this was after I have been scammed by various fake spell casters. I was introduced to Dr dodo a true Spell Caster. In less than 38 hours i saw wonders, my lover came back to me and my life got back just like a completed puzzle, and after 1month later we got married and it was just like a dream to me because i thought i had lost him forever. Thank you Dr. dodo for helping me but most of all, Your Honesty and Fast Accurate Results. EMAIL HIM FOR HELP: drdodotemple@ gmail. com or : through his number +(234) 8147766277.

    ARE THE THINGS DR dodo CAN THESE
    ALSO CURE.. GONORRHEA, HIV/AIDS , LOW SPERM COUNT, MENOPAUSE DISEASE, PREGNANCY PROBLEM, SHORT SIGHTEDNESS PROBLEM, Stroke, Bring back ex lover or wife/husband

    ReplyDelete
  29. A GREAT LOVE SPELL CASTER Dr,Uzoya HE SPECIALIZES IN RESTORING BROKEN MARRIAGES CONTACT HIM DIRECTLY ON HIS FACEBOOK PAGE (FB.ME/DRUZOYASPELLTEMPLE)..What else can i say rather than to thank Dr, Uzoya who God used to reunite my marriage. Each day of my life, i ask God to bless Dr,Uzoya for he has made my life complete by bringing back my husband to me and for this reason, i made a vow to my self that i will testify on the internet to let the world know that Dr, Uzoya is a God on Earth. My husband and i had a fight for three days which led to our divorce. On this faithful day, i came across a testimony of how Dr,Uzoya helped a lady in getting back her lover. So, i contacted him and explained to him and he told me that my days of sorrows are over that my husband will come back to me within 48 hours for he's going to prepare a spell for me. Could you believe it, my husband came home begging that he needs me back. Do you need any kind of help? Contact; (driraborspellcaster@gmail.com) OR call his phone number via +1(301)909-8775

    ReplyDelete
  30. This is the most wonderful thing i have ever experienced. I visited here online on the 17 june and i saw a marvelous testimony of Tracie Aldana from United States on the forum about the good works DR.tunde I never believed it, because have never heard anything about such miracle before. No body would have been able to convince me about it not until DR.tunde did a marvelous work for me that restored my mairage of 4 years by getting back my husband back i just as i read on the internet. I'm carolina peter I was truly shocked when my husband knelt down pleading for forgiveness to accept him back. I am really short of words to use to show my appreciation to DR.tunde For his a God sent to me and my entire family for divine restoration of my mariage. Contact him now for any kind of help via Email:babatundesolutioncentre1@gmail.com or you can contact his whatsapp mobile line +2348143581382 Also specialise in treating all kinds of illness, HERPES VIRUS, HIV AIDS, HEPATITIS B, CANCER, BRAIN DISEASE, PREGNANCY AND MORE.

    ReplyDelete
  31. The occurrences in my marriage wasn’t clear, I invested emotionally and I don’t see the positive impacts. I prayed for a husband that could remind me of the good he sees in me, a husband that can make me laugh and also boost my mood.
    The introduction of spell to my hunt saved my marriage. Dr. Wakina looked at the happenings from a different angle before solving the puzzle. He totally handled the event like the master he is. It was emotional for me seeing us reach beyond the invisible boundaries in our marriage.
    My husband is in support of this story, so I’m not compromising my integrity because the spell toss on us was a goodwill spell.
    “Share your story to liberate a relationship” was the word said to me by Dr. Wakina after he successfully completed a unity spell that brought us back after 9 months and 4 days. He found the enduring solution to my negative marital events via his email dr.wakinalovetemple@gmail.com. I bless the day I read the story that lead me to Dr. Wakina.

    ReplyDelete
  32. This is the most wonderful thing i have ever experienced. I visited here online on the 17 june and i saw a marvelous testimony of Tracie Aldana from United States on the forum about the good works DR.tunde I never believed it, because have never heard anything about such miracle before. No body would have been able to convince me about it not until DR.tunde did a marvelous work for me that restored my mairage of 4 years by getting back my husband back i just as i read on the internet. I'm carolina peter I was truly shocked when my husband knelt down pleading for forgiveness to accept him back. I am really short of words to use to show my appreciation to DR.tunde For his a God sent to me and my entire family for divine restoration of my mariage. Contact him now for any kind of help via Email:babatundesolutioncentre1@gmail.com or you can contact his whatsapp mobile line +2348143581382 Also specialise in treating all kinds of illness, HERPES VIRUS, HIV AIDS, HEPATITIS B, CANCER, BRAIN DISEASE, PREGNANCY AND MORE.

    ReplyDelete
  33. HOW TO GET YOUR EX BACK WITHOUT PANIC
    I am Saba from San Francisco. i want to testify about powerful Dr PATRICK for saving my relationship.I must be one of the few who don’t have a trouble showing and telling someone how grateful I am for making my relationship a melodious and smooth relationship. I have been trying to be more positive about my relationship with my boyfriend, it seems like it has taken forever. Another lady took my Fiancee away from me and they both relocated. i cried and searched for him to come back but he refused to take my calls nor return my messages. My friend introduced me to a powerful spell caster called Dr PATRICK who helped me cast a very powerful love spell that brought back my Fiancee in 24 hours. i was amazed and surprised when my fiancee called to beg me for forgiveness all because of the powerful love spell of Dr PATRICK. dont fail to Contact Dr PATRICK through the following means if you have any Marrital or Relationship problem. his contact information are

    Email: drpatrickspellcaster@gmail.com

    Mobile Number: +1 662 493 2087

    website: https://drpatrickspellcast.blogspot.com



    HOW TO GET YOUR EX BACK WITHOUT PANIC
    I am Saba from San Francisco. i want to testify about powerful Dr PATRICK for saving my relationship.I must be one of the few who don’t have a trouble showing and telling someone how grateful I am for making my relationship a melodious and smooth relationship. I have been trying to be more positive about my relationship with my boyfriend, it seems like it has taken forever. Another lady took my Fiancee away from me and they both relocated. i cried and searched for him to come back but he refused to take my calls nor return my messages. My friend introduced me to a powerful spell caster called Dr PATRICK who helped me cast a very powerful love spell that brought back my Fiancee in 24 hours. i was amazed and surprised when my fiancee called to beg me for forgiveness all because of the powerful love spell of Dr PATRICK. dont fail to Contact Dr PATRICK through the following means if you have any Marrital or Relationship problem. his contact information are

    Email: drpatrickspellcaster@gmail.com

    Mobile Number: +1 662 493 2087

    website: https://drpatrickspellcast.blogspot.com

    ReplyDelete
  34. i am very grateful sharing this great testimonies with you, The best thing that has ever happened in my life is how i win the lottery. I am a woman who believe that one day i will win the lottery.finally my dreams came through when i email Dr Alli . and tell him i need the lottery numbers. i have come a long way spending money on ticket just to make sure i win. But i never know that winning was so easy until the day i meant the spell caster online which so many people has talked about that he is very great in casting lottery spell, so i decide to give it a try.I contacted this man and he did a spell and he gave me the winning lottery numbers. But believe me when the draws were out i was among winners. i win 1.900.000 million Dollar. Dr. Alli truly you are the best, with these man you can will millions of money through lottery. i am so very happy to meet these man, i will forever be grateful to you. Email him for your own winning lottery numbers [ Allispellhelp1@gmail.com ]. OR Whatsapp or Call Via +2348100772528.

    ReplyDelete
  35. Contact Dr. Sam at supremespellcast@gmail.com if you want to get your Husband, Wife, Boyfriend and Girlfriend back now and regain your happiness.

    My name is Lisa Amy From USA. I am really pleased with this service Dr. Sam render to me. I am one of those people who said, “I’ll never call a psychic or a magician or whatever” to help me with my problems – least of all my love problems, but I reached the point where I knew I needed some guidance, and I’m so glad I found this man called Dr. Sam I never in a million years would have thought I’d be writing a letter like this, but when I nearly lost Andrew in one of our stupid, fights (he broke up with me), I thought I had lost everything.i cry all day and think that he will never come back to me again.I read a testimony by a man called Taylor Borg about him saying that how Dr. Sam helps him to get back is EX wife within 48 hours.I quickly email him. He did a (Lover spell) for me. 48 hours later, my (Ex-Boyfriend) came to me and apologized for the wrongs he did and promise never to do it again. Ever since then, everything has returned back to normal and our love is now stronger that he can't do without being with me. I and my boyfriend are living together happily again. Dr. Sam is the best online spell caster that is powerful and genuine. And when I was at my most desperate, he didn’t take advantage of me. You performed a very good service for a person in true need. I don’t know how you did it, or how this magic works, but all I know is, IT WORKS!! Andrew and i are happily back together, and I’ll always be grateful to Dr. Sam email him for any kinds of help is very capable and reliable for help. His blog> https://supremespellcast.blogspot.com

    Email him {Supremespellcast@gmail.com or Supremespellcast@yahoo.com}
    {Call/WhatsApp: +2347087462033}

    ReplyDelete
  36. My name is Gloria from Senegal, I came in here to confirm Hepatitis B. testimony that I red online some months ago. I red about how Dr. Iyabiye cured one Mr. Henry and I decided to give it a try to see if my sister could be healed or cured too, behold doctor Iyabiye is indeed a sent to help people out of this miserable disease. My sister was dying of hepatitis b her stomach was swollen and complained of pains and weaknesses in her body, I came across iyabiye’s contact online while checking about hepatitis. My sister got cured after she applied his medication for a month. I am here to thank Dr. Iyabiye and to encourage people to reach him for help through: iyabiyehealinghome@gmail.com Call/Whatsapp: +2348072229413

    ReplyDelete
  37. Get Your Lover Back Perfect Urgent Love Spell, Am Alexis scorpio from FL Usa All Thanks To Dr happy Who Brought Back The Love Of My Life. my girlfriend cheated on me and asked for a breakup. I don't believe at first i try to get back with her but all she told me was she’s with someone else. that she is no longer interested in marrying me at that point i was heart broken coz i love my girlfriend so much that i could not let go off her all of a sudden she left me, i really love her and never can imagine my life without her. not until i came across a powerful real spell caster Dr happy who promise me 24hours urgent love spell to get back with my girlfriend, good twenty-four {24} hours. hmm-mm, it was a good night time at 11:pm within the days that Dr happy told me that my girlfriend will be back, at first i heard the bell rings getting close to my door i heard someone saying honey!!!, it sound familiar i opened the door and i saw my girlfriend standing and weeping in front of me. i was not surprised because its all i have been praying for her to come back home. Guess what 2 weeks after she noticed her system and her body temperature was changed and i took her to clinic for check up and the doctor told me that there is life in her which means she was pregnant i really wants to use this opportunity to thanks Dr happy so much and the love page that directed me to Dr happy if you have any problem getting your ex back, or predicament that is worse or exactly like this you have been into, contact Dr happy on happylovespell2@gmail.com
    Whatsapp/cal +2348133873774
    visit his website...happylovespell2.webnode.com/
    about his love spell Blogs site.http://happylovespel.blogspot.com.ng/

    ReplyDelete
  38. I was diagnose of Chronic Obstructive Pulmonary Disease (COPD) 2 years ago, my family doctor told me there is no cure but it can only be controled. Ever since then i have been running heather scepter looking for a solution to my problem because i was going to lose my Job if i dont get better, until i ran into this online testimony of Mrs Alicia who have been cured of COPD by this herbal clinic in Africa (Astoria Herbal Clinic), i immediately contacted them via their email (azentemple@gmailcom), to God be thy glory i bought this COPD herbal remedy and was cured within 3 weeks of usage, at last this is a break through for all COPD patients contact this herbal clinic via their email ( azentemple@gmailcom) or whasapp +2349038299828 He also have the herb to cure:
    (1) ALS,
    (2) DIABETES,
    (3) HIV&AIDS,
    (4) URINARY TRACT INFECTION,
    (5) CANCER,
    (6) HERPES

    ReplyDelete
  39. I was diagnose of Chronic Obstructive Pulmonary Disease (COPD) 2 years ago, my family doctor told me there is no cure but it can only be controled. Ever since then i have been running heather scepter looking for a solution to my problem because i was going to lose my Job if i dont get better, until i ran into this online testimony of Mrs Alicia who have been cured of COPD by this herbal clinic in Africa (Astoria Herbal Clinic), i immediately contacted them via their email (azentemple@gmailcom), to God be thy glory i bought this COPD herbal remedy and was cured within 3 weeks of usage, at last this is a break through for all COPD patients contact this herbal clinic via their email ( azentemple@gmailcom) or whasapp +2349038299828 He also have the herb to cure:
    (1) ALS,
    (2) DIABETES,
    (3) HIV&AIDS,
    (4) URINARY TRACT INFECTION,
    (5) CANCER,
    (6) HERPES

    ReplyDelete
  40. Get Your Lover Back Perfect Urgent Love Spell, Am Alexis scorpio from FL Usa All Thanks To Dr happy Who Brought Back The Love Of My Life. my girlfriend cheated on me and asked for a breakup. I don't believe at first i try to get back with her but all she told me was she’s with someone else. that she is no longer interested in marrying me at that point i was heart broken coz i love my girlfriend so much that i could not let go off her all of a sudden she left me, i really love her and never can imagine my life without her. not until i came across a powerful real spell caster Dr happy who promise me 24hours urgent love spell to get back with my girlfriend, good twenty-four {24} hours. hmm-mm, it was a good night time at 11:pm within the days that Dr happy told me that my girlfriend will be back, at first i heard the bell rings getting close to my door i heard someone saying honey!!!, it sound familiar i opened the door and i saw my girlfriend standing and weeping in front of me. i was not surprised because its all i have been praying for her to come back home. Guess what 2 weeks after she noticed her system and her body temperature was changed and i took her to clinic for check up and the doctor told me that there is life in her which means she was pregnant i really wants to use this opportunity to thanks Dr happy so much and the love page that directed me to Dr happy if you have any problem getting your ex back, or predicament that is worse or exactly like this you have been into, contact Dr happy on happylovespell2@gmail.com
    Whatsapp/cal +2348133873774
    visit his website...happylovespell2.webnode.com/
    about his love spell Blogs site.http://happylovespel.blogspot.com.ng/

    ReplyDelete
  41. This is the most wonderful thing i have ever experienced. I visited here online on the 17 june and i saw a marvelous testimony of Tracie Aldana from United States on the forum about the good works DR.tunde I never believed it, because have never heard anything about such miracle before. No body would have been able to convince me about it not until DR.tunde did a marvelous work for me that restored my mairage of 4 years by getting back my husband back i just as i read on the internet. I'm carolina peter I was truly shocked when my husband knelt down pleading for forgiveness to accept him back. I am really short of words to use to show my appreciation to DR.tunde For his a God sent to me and my entire family for divine restoration of my mariage. Contact him now for any kind of help via Email:babatundesolutioncentre1@gmail.com or you can contact his whatsapp mobile line +2348143581382 Also specialise in treating all kinds of illness, HERPES VIRUS, HEPATITIS B, CANCER, BRAIN DISEASE, PREGNANCY, DIABETIES AND MORE.

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  42. HOW I GOT MY EX HUSBAND BACK WITH THE HELP OF REAL AND EFFECTIVE SPELL FROM DR Osasu
    My name is Rosemarie Williams, I never thought I will smile again, My husband left me with two kids for one year, All effort to bring him back failed I thought I'm not going to see him again not until I met a lady called Jesse who told me about a spell caster called Dr. Osasu , She gave me his email address and mobile number and I contacted him and he assured me that within 48hours my husband will come back to me, In less than 48hours my husband came back started begging for forgiveness saying it is the devils work, so I'm still surprise till now about this miracle,i couldn't conceive but as soon as the spell was cast,i became pregnant and gave birth to my third child,if you need any assistance from him you can contact him via:email: drosasu25@gmail.com Or WhatsApp or call him now: +2347064365391

    ReplyDelete
  43. HOW I GOT MY EX HUSBAND BACK WITH THE HELP OF REAL AND EFFECTIVE SPELL FROM DR Osasu
    My name is Rosemarie Williams, I never thought I will smile again, My husband left me with two kids for one year, All effort to bring him back failed I thought I'm not going to see him again not until I met a lady called Jesse who told me about a spell caster called Dr. Osasu , She gave me his email address and mobile number and I contacted him and he assured me that within 48hours my husband will come back to me, In less than 48hours my husband came back started begging for forgiveness saying it is the devils work, so I'm still surprise till now about this miracle,i couldn't conceive but as soon as the spell was cast,i became pregnant and gave birth to my third child,if you need any assistance from him you can contact him via:email: drosasu25@gmail.com Or WhatsApp or call him now: +2347064365391

    ReplyDelete