Friday, November 11, 2016

LeetCode 458 - Poor Pigs


https://leetcode.com/problems/poor-pigs/
There are 1000 buckets, one and only one of them contains poison, the rest are filled with water. They all look the same. If a pig drinks that poison it will die within 15 minutes. What is the minimum amount of pigs you need to figure out which bucket contains the poison within one hour.
Answer this question, and write an algorithm for the follow-up general case.
Follow-up:


If there are n buckets and a pig drinking poison will die within m minutes, how many pigs (x) you need to figure out the "poison" bucket within p minutes? There is exact one bucket with poison.
https://discuss.leetcode.com/topic/66856/major-flaw-in-current-algorithm-fixed
The basic case (and presumably a significant amount more) are incorrect as more efficient solutions can be generated if we do not use base 2.
Assume for now we have: [buckets, 15, 60]
Per pig, we have 5 states that we can detect: death on 15, 30, 45, 60, or life.
Assume for the moment we have 5 buckets and one pig with the above timings.
Represent each bucket as an integer in base 5 (0-indexed).
Pig drinks from bucket 0 at 0 minutes, bucket 1 at time 15 etc...
If the pig is still alive, by elimination, we know that bucket 4 would be poisonous.
This should be trivial to see.
We now increase the number of buckets to 25, we have ceil(log_5(25)) = 2 pigs
Bucket 25 is labelled 44, bucket 1 is labelled 00.
At time t, have the first pig drink from the buckets where the first digit is t, have the second pig drink when the second digit is t. As each pig represents a single digit in base 5, and 25 consists of two base 5 digits, two pigs are necessary. We can determine the value of each digit based on the time of death of each pig. An overlap of the death of two pigs gives us the information of two digits, so nothing is lost.
Example: 18 is poisonous (33 in base 5).
On t=3, one pig is drinking [15,16,17,18,19] and the other is drinking [3,8,13,18,23]. We can see that if both die, the intersection of the sets yields 18.
Given the default input: 1000, 15, 60
The above yields the minimum number of pigs = ceil(log_5(1000)) = 5 pigs.
The OJ currently returns 8 pigs, which is significantly higher.
    public int poorPigs(int buckets, int minutesToDie, int minutesToTest) {
        if (buckets--==1){
            return 0;
        }
        int base=minutesToTest/minutesToDie+1;
        int count=0;
        while (buckets>0){
            buckets/=base;
            count++;
        }
        return count;
    }
https://discuss.leetcode.com/topic/67176/math-problem-java-ac-code-with-brief-explanations-11-09-2016
The table below shows how it works, n denotes the pig, and s is associated with each status of the pig.
for each pig, there is t+1 status, which is death in 1st, 2nd, 3rd, 4th time slot and live eventually.
Therefore the question is asking for how many pigs can denote 1000 in 5-nary number system.
=======Table=======
n1 n2 n3 n4 n5
s1 1 0 0 1 0
s2 0 0 0 0 1
s3 0 1 0 0 0
s4 0 0 0 0 0
s5 0 0 1 0 0


    In this case, the number is 13012 in 5-nary number system.
    public int poorPigs(int buckets, int minutesToDie, int minutesToTest) {
            int status = minutesToTest/minutesToDie + 1;
            int num_of_pig = 0;
            while(Math.pow(status, num_of_pig) < buckets) {num_of_pig++;}
            return num_of_pig;
        }
    

    http://bookshadow.com/weblog/2016/11/08/leetcode-poor-pigs/
    数学题(Mathematics)
    令r = p / m,表示在规定时间内可以做多少轮“试验”。
    假设有3头猪,计算可以确定的最大桶数。
    首先考虑只能做1轮试验的情形:
    0 1 2 3 4 5 6 7 (桶序号)
    0 0 0 0 1 1 1 1 (第1头猪饮用:4,5,6,7)
    0 0 1 1 0 0 1 1 (第2头猪饮用:2,3,6,7)
    0 1 0 1 0 1 0 1 (第3头猪饮用:1,3,5,7)
    可能的试验结果如下(0表示幸存, 1表示死亡):
    0 1 2 (猪序号)
    0 0 0 (毒药为0号桶,0、1、2号猪均幸存)
    0 0 1 (毒药为1号桶,2号猪死亡)
    0 1 0 (毒药为2号桶,1号猪死亡)
    0 1 1 (毒药为3号桶,1、2号猪死亡)
    1 0 0 (毒药为4号桶,0号猪死亡)
    1 0 1 (毒药为5号桶,0、2号猪死亡)
    1 1 0 (毒药为6号桶,0、1号猪死亡)
    1 1 1 (毒药为7号桶,0、1、2号猪均死亡)
    因此3头猪做1轮试验可以确定的最大桶数为pow(2, 3) = 8
    接下来考虑做2轮试验的情形:
    与只做一轮试验类似,依然采用二进制分组的形式。
    不同之处在于,每一个分组内可以包含若干个桶,第1轮试验确定分组,第2轮试验确定具体的桶。
    0 1 2 3 4 5 6 7 (组序号)
    0 0 0 0 1 1 1 1 (第1头猪饮用:4,5,6,7)
    0 0 1 1 0 0 1 1 (第2头猪饮用:2,3,6,7)
    0 1 0 1 0 1 0 1 (第3头猪饮用:1,3,5,7)
    8 4 4 2 4 2 2 1 (桶数量)
    各分组包含的桶数量根据做完第1轮试验后猪的幸存情况确定。
    幸存猪数     桶数量     可能的情况数
       3           8             1
       2           4             3
       1           2             3
       0           1             1
    合计: 1 * 8 + 3 * 4 + 3 * 2 + 1 * 1 = 27
    因此3头猪做2轮试验可以确定的最大桶数为pow(3, 3) = 27
    实际上1,3,3,1是二项式系数,8,4,2,1是2的幂
    由此可以归纳出公式:
    buckets = ∑( C(n, i) * pow(r, i) ) 
    
    其中 i∈ [0, n],C是二项式系数,r为做试验的轮数,n表示猪的数量
    化简上式:
    buckets = pow(r + 1, n)
    最终得:
    n = log(buckets, r + 1)
    def poorPigs(self, buckets, minutesToDie, minutesToTest): """ :type buckets: int :type minutesToDie: int :type minutesToTest: int :rtype: int """ return int(math.ceil(math.log(buckets, 1 + minutesToTest / minutesToDie)))



    No comments:

    Post a Comment

    Labels

    GeeksforGeeks (1107) LeetCode (993) Algorithm (795) Review (766) to-do (633) LeetCode - Review (514) Classic Algorithm (324) Dynamic Programming (293) Classic Interview (288) Google Interview (242) Tree (145) POJ (139) Difficult Algorithm (132) LeetCode - Phone (127) EPI (125) Different Solutions (120) Bit Algorithms (118) Lintcode (113) Cracking Coding Interview (110) Smart Algorithm (109) Math (107) HackerRank (89) Binary Tree (82) Binary Search (81) Graph Algorithm (74) Greedy Algorithm (72) DFS (67) LeetCode - Extended (62) Interview Corner (61) Stack (60) List (58) Advanced Data Structure (56) BFS (54) Codility (54) ComProGuide (52) Algorithm Interview (50) Geometry Algorithm (48) Binary Search Tree (46) USACO (46) Trie (45) Mathematical Algorithm (42) ACM-ICPC (41) Interval (41) Data Structure (40) Knapsack (40) Space Optimization (40) Jobdu (39) LeetCode Hard (39) Recursive Algorithm (39) Matrix (38) String Algorithm (38) Backtracking (36) Codeforces (36) Introduction to Algorithms (36) Must Known (36) Beauty of Programming (35) Sort (35) Union-Find (34) Array (33) prismoskills (33) Segment Tree (32) Sliding Window (32) Data Structure Design (31) HDU (31) Google Code Jam (30) Permutation (30) Puzzles (30) Array O(N) (29) Company-Airbnb (29) Company-Zenefits (28) Microsoft 100 - July (28) Palindrome (28) to-do-must (28) Priority Queue (27) Random (27) Graph (26) Company - LinkedIn (25) GeeksQuiz (25) Logic Thinking (25) Pre-Sort (25) hihocoder (25) Queue (24) Company-Facebook (23) High Frequency (23) Post-Order Traverse (23) TopCoder (23) Algorithm Game (22) Bisection Method (22) Hash (22) Binary Indexed Trees (21) DFS + Review (21) Lintcode - Review (21) Brain Teaser (20) CareerCup (20) Company - Twitter (20) Merge Sort (20) O(N) (20) Follow Up (19) Time Complexity (19) Two Pointers (19) UVA (19) Ordered Stack (18) Probabilities (18) Company-Uber (17) Game Theory (17) Topological Sort (17) Codercareer (16) Heap (16) Shortest Path (16) String Search (16) Tree Traversal (16) itint5 (16) Difficult (15) Iterator (15) BST (14) Number (14) Number Theory (14) Amazon Interview (13) Basic Algorithm (13) Codechef (13) Euclidean GCD (13) KMP (13) Long Increasing Sequence(LIS) (13) Majority (13) mitbbs (13) Combination (12) Computational Geometry (12) LeetCode - Classic (12) Modify Tree (12) Reconstruct Tree (12) Reservoir Sampling (12) Reverse Thinking (12) 尺取法 (12) AOJ (11) DFS+Backtracking (11) Fast Power Algorithm (11) Graph DFS (11) LCA (11) LeetCode - DFS (11) Miscs (11) Princeton (11) Proof (11) Tree DP (11) X Sum (11) 挑战程序设计竞赛 (11) Bisection (10) Bucket Sort (10) Coin Change (10) Company - Microsoft (10) DFS+Cache (10) Facebook Hacker Cup (10) HackerRank Easy (10) O(1) Space (10) Rolling Hash (10) SPOJ (10) Theory (10) Tutorialhorizon (10) DP-Multiple Relation (9) DP-Space Optimization (9) Divide and Conquer (9) Kadane - Extended (9) Mathblog (9) Max-Min Flow (9) Prefix Sum (9) Quick Sort (9) Simulation (9) Stack Overflow (9) Stock (9) System Design (9) TreeMap (9) Use XOR (9) Book Notes (8) Bottom-Up (8) Company-Amazon (8) DFS+BFS (8) LeetCode - DP (8) Left and Right Array (8) Linked List (8) Longest Common Subsequence(LCS) (8) Prime (8) Suffix Tree (8) Tech-Queries (8) Traversal Once (8) 穷竭搜索 (8) Algorithm Problem List (7) Expression (7) Facebook Interview (7) Fibonacci Numbers (7) Game Nim (7) Graph BFS (7) HackerRank Difficult (7) Hackerearth (7) Interval Tree (7) Inversion (7) Kadane’s Algorithm (7) Level Order Traversal (7) Math-Divisible (7) Probability DP (7) Quick Select (7) Radix Sort (7) n00tc0d3r (7) 蓝桥杯 (7) Catalan Number (6) Classic Data Structure Impl (6) DFS+DP (6) DP - Tree (6) DP-Print Solution (6) Dijkstra (6) Dutch Flag (6) How To (6) Interviewstreet (6) Knapsack - MultiplePack (6) Manacher (6) Minimum Spanning Tree (6) Morris Traversal (6) Multiple Data Structures (6) One Pass (6) Programming Pearls (6) Pruning (6) Rabin-Karp (6) Randomized Algorithms (6) Sampling (6) Schedule (6) Stream (6) Suffix Array (6) Threaded (6) TreeSet (6) Xpost (6) reddit (6) AI (5) Algorithm - Brain Teaser (5) Art Of Programming-July (5) Big Data (5) Brute Force (5) Code Kata (5) Codility-lessons (5) Coding (5) Company - WMware (5) Convex Hull (5) Crazyforcode (5) Cycle (5) DP-Include vs Exclude (5) Fast Slow Pointers (5) Find Rule (5) Graph Cycle (5) Hash Strategy (5) Immutability (5) Java (5) Matrix Chain Multiplication (5) Maze (5) Microsoft Interview (5) Pre-Sum (5) Quadtrees (5) Quick Partition (5) Quora (5) SPFA(Shortest Path Faster Algorithm) (5) Subarray Sum (5) Sudoku (5) Sweep Line (5) Word Search (5) jiuzhang (5) 单调栈 (5) 树形DP (5) 1point3acres (4) Abbreviation (4) Anagram (4) Anagrams (4) Approximate Algorithm (4) Backtracking-Include vs Exclude (4) Brute Force - Enumeration (4) Chess Game (4) Consistent Hash (4) Distributed (4) Eulerian Cycle (4) Flood fill (4) Graph-Classic (4) HackerRank AI (4) Histogram (4) Kadane Max Sum (4) Knapsack - Mixed (4) Knapsack - Unbounded (4) LeetCode - Recursive (4) LeetCode - TODO (4) MST (4) MinMax (4) N Queens (4) Nerd Paradise (4) Parallel Algorithm (4) Practical Algorithm (4) Probability (4) Programcreek (4) Spell Checker (4) Stock Maximize (4) Subset Sum (4) Subsets (4) Symbol Table (4) Triangle (4) Water Jug (4) algnotes (4) fgdsb (4) to-do-2 (4) 最大化最小值 (4) A Star (3) Algorithm - How To (3) Algorithm Design (3) B Tree (3) Big Data Algorithm (3) Caterpillar Method (3) Coins (3) Company - Groupon (3) Company - Indeed (3) Cumulative Sum (3) DP-Fill by Length (3) DP-Two Variables (3) Dedup (3) Dequeue (3) Dropbox (3) Easy (3) Finite Automata (3) Github (3) GoLang (3) Graph - Bipartite (3) Include vs Exclude (3) Joseph (3) Jump Game (3) K (3) Knapsack-多重背包 (3) LeetCode - Bit (3) Linked List Merge Sort (3) LogN (3) Master Theorem (3) Min Cost Flow (3) Minesweeper (3) Missing Numbers (3) NP Hard (3) O(N) Hard (3) Online Algorithm (3) Pascal's Triangle (3) Pattern Match (3) Project Euler (3) Rectangle (3) Scala (3) SegmentFault (3) Shuffle (3) Sieve of Eratosthenes (3) Stack - Smart (3) State Machine (3) Subtree (3) Transform Tree (3) Trie + DFS (3) Two Pointers Window (3) Warshall Floyd (3) With Random Pointer (3) Word Ladder (3) bookkeeping (3) codebytes (3) Activity Selection Problem (2) Advanced Algorithm (2) AnAlgorithmADay (2) Application of Algorithm (2) Array Merge (2) BOJ (2) BT - Path Sum (2) Balanced Binary Search Tree (2) Bellman Ford (2) Binary Search - Smart (2) Binomial Coefficient (2) Bit Counting (2) Bit Mask (2) Bit-Difficult (2) Bloom Filter (2) Book Coding Interview (2) Branch and Bound Method (2) Clock (2) Codesays (2) Company - Baidu (2) Company-Snapchat (2) Complete Binary Tree (2) DFS+BFS, Flood Fill (2) DP - DFS (2) DP-3D Table (2) DP-Classical (2) DP-Output Solution (2) DP-Slide Window Gap (2) DP-i-k-j (2) DP-树形 (2) Distributed Algorithms (2) Divide and Conqure (2) Doubly Linked List (2) Edit Distance (2) Factor (2) Forward && Backward Scan (2) GoHired (2) Graham Scan (2) Graph BFS+DFS (2) Graph Coloring (2) Graph-Cut Vertices (2) Hamiltonian Cycle (2) Huffman Tree (2) In-order Traverse (2) Include or Exclude Last Element (2) Information Retrieval (2) Interview - Linkedin (2) Invariant (2) Islands (2) Linked Interview (2) Linked List Sort (2) Longest SubArray (2) Lucene-Solr (2) Math-Remainder Queue (2) Matrix Power (2) Median (2) Minimum Vertex Cover (2) Negative All Values (2) Number Each Digit (2) Numerical Method (2) Object Design (2) Order Statistic Tree (2) Parent-Only Tree (2) Parentheses (2) Parser (2) Peak (2) Programming (2) Range Minimum Query (2) Regular Expression (2) Return Multiple Values (2) Reuse Forward Backward (2) Robot (2) Rosettacode (2) Scan from right (2) Search (2) SimHash (2) Simple Algorithm (2) Skyline (2) Spatial Index (2) Strongly Connected Components (2) Summary (2) TV (2) Tile (2) Traversal From End (2) Tree Sum (2) Tree Traversal Return Multiple Values (2) Tree Without Tree Predefined (2) Word Break (2) Word Graph (2) Word Trie (2) Yahoo Interview (2) Young Tableau (2) 剑指Offer (2) 数位DP (2) 1-X (1) 51Nod (1) Akka (1) Algorithm - New (1) Algorithm Series (1) Algorithms Part I (1) Analysis of Algorithm (1) Array-Element Index Negative (1) Array-Rearrange (1) Augmented BST (1) Auxiliary Array (1) Auxiliary Array: Inc&Dec (1) BACK (1) BK-Tree (1) BZOJ (1) Basic (1) Bayes (1) Beauty of Math (1) Big Integer (1) Big Number (1) Binary (1) Binary Sarch Tree (1) Binary String (1) Binary Tree Variant (1) Bipartite (1) Bit-Missing Number (1) BitMap (1) BitMap index (1) BitSet (1) Bug Free Code (1) BuildIt (1) C/C++ (1) CC Interview (1) Cache (1) Calculate Height at Same Recusrion (1) Cartesian tree (1) Check Tree Property (1) Chinese (1) Circular Buffer (1) Cloest (1) Clone (1) Code Quality (1) Codesolutiony (1) Company - Alibaba (1) Company - Palantir (1) Company - WalmartLabs (1) Company-Apple (1) Company-Epic (1) Company-Salesforce (1) Company-Yelp (1) Compression Algorithm (1) Concurrency (1) Cont Improvement (1) Convert BST to DLL (1) Convert DLL to BST (1) Custom Sort (1) Cyclic Replacement (1) DFS-Matrix (1) DP - Probability (1) DP Fill Diagonal First (1) DP-Difficult (1) DP-End with 0 or 1 (1) DP-Fill Diagonal First (1) DP-Graph (1) DP-Left and Right Array (1) DP-MaxMin (1) DP-Memoization (1) DP-Node All Possibilities (1) DP-Optimization (1) DP-Preserve Previous Value (1) DP-Print All Solution (1) Database (1) Detect Negative Cycle (1) Diagonal (1) Directed Graph (1) Do Two Things at Same Recusrion (1) Domino (1) Dr Dobb's (1) Duplicate (1) Equal probability (1) External Sort (1) FST (1) Failure Function (1) Fraction (1) Front End Pointers (1) Funny (1) Fuzzy String Search (1) Game (1) Generating Function (1) Generation (1) Genetic algorithm (1) GeoHash (1) Geometry - Orientation (1) Google APAC (1) Graph But No Graph (1) Graph Transpose (1) Graph Traversal (1) Graph-Coloring (1) Graph-Longest Path (1) Gray Code (1) HOJ (1) Hanoi (1) Hard Algorithm (1) How Hash (1) How to Test (1) Improve It (1) In Place (1) Inorder-Reverse Inorder Traverse Simultaneously (1) Interpolation search (1) Interview (1) Interview - Facebook (1) Isomorphic (1) JDK8 (1) K Dimensional Tree (1) Knapsack - Fractional (1) Knapsack - ZeroOnePack (1) Knight (1) Knuth Shuffle (1) Kosaraju’s algorithm (1) Kruskal (1) Kth Element (1) Least Common Ancestor (1) LeetCode - Binary Tree (1) LeetCode - Coding (1) LeetCode - Detail (1) LeetCode - Related (1) Linked List Reverse (1) Linkedin (1) Linkedin Interview (1) Local MinMax (1) Logic Pattern (1) Longest Common Subsequence (1) Longest Common Substring (1) Longest Prefix Suffix(LPS) (1) Machine Learning (1) Maintain State (1) Manhattan Distance (1) Map && Reverse Map (1) Math - Induction (1) Math-Multiply (1) Math-Sum Of Digits (1) Matrix - O(N+M) (1) Matrix BFS (1) Matrix Graph (1) Matrix Search (1) Matrix+DP (1) Matrix-Rotate (1) Max Min So Far (1) Memory-Efficient (1) MinHash (1) MinMax Heap (1) Monotone Queue (1) Monto Carlo (1) Multi-End BFS (1) Multi-Reverse (1) Multiple DFS (1) Multiple Tasks (1) Next Element (1) Next Successor (1) Offline Algorithm (1) PAT (1) Parenthesis (1) Partition (1) Path Finding (1) Patience Sort (1) Persistent (1) Pigeon Hole Principle (1) Power Set (1) Pratical Algorithm (1) PreProcess (1) Probabilistic Data Structure (1) Python (1) Queue & Stack (1) RSA (1) Ranking (1) Rddles (1) ReHash (1) Realtime (1) Recurrence Relation (1) Recursive DFS (1) Recursive to Iterative (1) Red-Black Tree (1) Region (1) Resources (1) Reverse Inorder Traversal (1) Robin (1) Selection (1) Self Balancing BST (1) Similarity (1) Sort && Binary Search (1) Square (1) Streaming Algorithm (1) String Algorithm. Symbol Table (1) String DP (1) String Distance (1) SubMatrix (1) Subsequence (1) System of Difference Constraints(差分约束系统) (1) TSP (1) Ternary Search Tree (1) Test (1) Test Cases (1) Thread (1) TimSort (1) Top-Down (1) Tournament (1) Tournament Tree (1) Transform Tree in Place (1) Tree Diameter (1) Tree Rotate (1) Trie and Heap (1) Trie vs Hash (1) Trie vs HashMap (1) Triplet (1) Two Data Structures (1) Two Stacks (1) USACO - Classical (1) USACO - Problems (1) UyHiP (1) Valid Tree (1) Vector (1) Virtual Matrix (1) Wiggle Sort (1) Wikipedia (1) ZOJ (1) ZigZag (1) baozitraining (1) codevs (1) cos126 (1) javabeat (1) jum (1) namic Programming (1) sqrt(N) (1) 两次dijkstra (1) 九度 (1) 二进制枚举 (1) 夹逼法 (1) 归一化 (1) 折半枚举 (1) 枚举 (1) 状态压缩DP (1) 男人八题 (1) 英雄会 (1) 逆向思维 (1)

    Popular Posts