. . February 3rd, 2011 Biostatistics 615/815 - Lecture 9 Hyun Min Kang February 3rd, 2011 Hyun Min Kang Dynamic Programming Biostatistics 615/815 Lecture 9: . . . . . . Summary . . Edit Distance MTP Fibonacci Introduction . . . . . . . . 1 / 34 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . good performance . Key components . . . . . . . . Hash function h x key mapping key onto smaller ’addressible’ space H Total required memory is the possible number of hash values Good hash function minimize the possibility of key collisions Collision-resolution strategy, when h k h k . Hyun Min Kang Biostatistics 615/815 - Lecture 9 February 3rd, 2011 . . . . . . . . . . . . Introduction Fibonacci MTP Edit Distance . Summary Recap: Hash Tables . Key features . . . . 2 / 34 . . . . . . . . . . . . . . . . . . . . . . . . . . . . • Θ(1) complexity for Insert , Search , and Remove • Requires large memory space than the actual content for maintainng • But uses much smaller memory than direct-addres tables
. . . . . good performance . Key components . . . . . . . . Hyun Min Kang Biostatistics 615/815 - Lecture 9 February 3rd, 2011 . . . MTP . . . . . . . . Introduction Fibonacci . 2 / 34 Edit Distance Recap: Hash Tables . . Summary Key features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • Θ(1) complexity for Insert , Search , and Remove • Requires large memory space than the actual content for maintainng • But uses much smaller memory than direct-addres tables • Hash function • h ( x . key ) mapping key onto smaller ’addressible’ space H • Total required memory is the possible number of hash values • Good hash function minimize the possibility of key collisions • Collision-resolution strategy, when h ( k 1 ) = h ( k 2 ) .
. MTP February 3rd, 2011 Biostatistics 615/815 - Lecture 9 Hyun Min Kang Recap: Illustration of ChainedHash Summary . Edit Distance 3 / 34 . Fibonacci Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . Double Hashing . . . . . . . . Hyun Min Kang Biostatistics 615/815 - Lecture 9 February 3rd, 2011 . . . MTP . . . . . . . . Introduction . Fibonacci Edit Distance Summary Probing strategies . Recap : Open hash 4 / 34 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • Linear probing • Quadratic probing • Double hashing • h ( k , i ) = ( h 1 ( k ) + ih 2 ( k )) mod m • The probe sequence depends in two ways upon k . • For example, h 1 ( k ) = k mod m , h 2 ( k ) = 1 + ( k mod m ′ ) • Avoid clustering problem • Performance close to ideal scheme of uniform hashing.
. Today February 3rd, 2011 Biostatistics 615/815 - Lecture 9 Hyun Min Kang . . . . . . . . Dynamic Programming . . Summary Introduction . . . . . . . . . MTP Fibonacci Edit Distance 5 / 34 . . . . . . . . . . . . . . . . . . . . . . . . . . . . • Fibonacci numbers • Manhattan tourist problems • Edit distance problem
. . February 3rd, 2011 Biostatistics 615/815 - Lecture 9 Hyun Min Kang until they become trivial. These algorithms divide a problem into smaller and disjoint subproblems . . . . . . . . . Good examples of divide and conquer algorithms Recap: Divide and conquer algorithms Introduction . . . . . . . . Summary 6 / 34 MTP Fibonacci Edit Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • TowerOfHanoi • MergeSort • QuickSort • BinarySearchTree algorithms
int fibonacci(int n) { if ( n < 2 ) return n; else return fibonacci(n-1)+fibonacci(n-2); } . . . . . A recursive implementation of fibonacci numbers . . . . . . . . Hyun Min Kang Biostatistics 615/815 - Lecture 9 February 3rd, 2011 . 7 / 34 . . . . . . . . . . Introduction Fibonacci MTP Edit Distance . Summary . A divide-and-conquer algorithms for Fibonacci numbers Fibonacci numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F n − 1 + F n − 2 n > 1 F n = 1 n = 1 0 n = 0
. . . . . . . A recursive implementation of fibonacci numbers . . . . . . . . Hyun Min Kang Biostatistics 615/815 - Lecture 9 February 3rd, 2011 . 7 / 34 . . . . . . . . . . Introduction Fibonacci MTP . Edit Distance Summary . Fibonacci numbers . A divide-and-conquer algorithms for Fibonacci numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . F n − 1 + F n − 2 n > 1 F n = 1 n = 1 0 n = 0 int fibonacci(int n) { if ( n < 2 ) return n; else return fibonacci(n-1)+fibonacci(n-2); }
. Performance of recursive Fibonacci February 3rd, 2011 Biostatistics 615/815 - Lecture 9 Hyun Min Kang . . . . . . . . Computational time . . Summary Introduction . . . . . . . . . MTP Fibonacci Edit Distance 8 / 34 . . . . . . . . . . . . . . . . . . . . . . . . . . . . • 4.4 seconds for calculating F 40 • 49 seconds for calculating F 45 • ∞ seconds for calculating F 100 !
. MTP February 3rd, 2011 Biostatistics 615/815 - Lecture 9 Hyun Min Kang What is happening in the recursive Fibonacci Summary . Edit Distance 9 / 34 . Fibonacci Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. Edit Distance February 3rd, 2011 Biostatistics 615/815 - Lecture 9 Hyun Min Kang The time complexity is exponential . Time complexity of redundant Fibonacci Summary . 10 / 34 MTP Fibonacci Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T ( n ) = T ( n − 1) + T ( n − 2) T (1) = 1 T (0) = 1 T ( n ) = F n +1
. MTP February 3rd, 2011 Biostatistics 615/815 - Lecture 9 Hyun Min Kang . Summary . Edit Distance A non-redundant Fibonacci 11 / 34 Fibonacci . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . int fibonacci(int n) { int* fibs = new int[n+1]; fibs[0] = 0; fibs[1] = 1; for(int i=2; i <= n; ++i) { fibs[i] = fibs[i-1]+fibs[i-2]; } int ret = fibs[n]; delete [] fibs; return ret; }
. . February 3rd, 2011 Biostatistics 615/815 - Lecture 9 Hyun Min Kang Key idea in non-redundant Fibonacci Summary . Edit Distance MTP Fibonacci Introduction . . . . . . . . 12 / 34 . . . . . . . . . . . . . . . . . . . . . . . . . . . . • Each F n will be reused to calculate F n +1 and F n +2 • Store F n into an array so that we don’t have to recalculate it
. MTP February 3rd, 2011 Biostatistics 615/815 - Lecture 9 Hyun Min Kang . Summary . Edit Distance A recursive, but non-redundant Fibonacci 13 / 34 Fibonacci . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . int fibonacci(int* fibs, int n) { if ( fibs[n] > 0 ) { return fibs[n]; // reuse stored solution if available } else if ( n < 2 ) { return n; // terminal condition } fibs[n] = fibonacci(n-1) + fibonacci(n-2); // store the solution once computed return fibs[n]; }
. . . . . . . . . . According to wikipedia... ”The word ’dynamic’ was chosen because it sounded impressive, not because how the method works” . Examples of dynamic programming . . . . . . . . Shortest path finding algorithms DNA sequence alignment Hidden markov models Hyun Min Kang Biostatistics 615/815 - Lecture 9 February 3rd, 2011 Why dynamic programming? 14 / 34 . Dynamic programming . . . . . . . . Introduction Fibonacci MTP Edit Distance . Summary . Key components of dynamic programing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • Problems that can be divided into subproblems • Overlapping subproblems - subproblems share subsubproblems • Solves each subsubproblem just once and then saves its answer
. . . . . . . . . . According to wikipedia... ”The word ’dynamic’ was chosen because it sounded impressive, not because how the method works” . Examples of dynamic programming . . . . . . . . Shortest path finding algorithms DNA sequence alignment Hidden markov models Hyun Min Kang Biostatistics 615/815 - Lecture 9 February 3rd, 2011 Why dynamic programming? 14 / 34 . Dynamic programming . . . . . . . . Introduction Fibonacci MTP Edit Distance . Summary . Key components of dynamic programing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • Problems that can be divided into subproblems • Overlapping subproblems - subproblems share subsubproblems • Solves each subsubproblem just once and then saves its answer
. . Why dynamic programming? . . . . . . . . According to wikipedia... ”The word ’dynamic’ was chosen because it sounded impressive, not because how the method works” Examples of dynamic programming . . . . . . . . . Hyun Min Kang Biostatistics 615/815 - Lecture 9 February 3rd, 2011 . 14 / 34 Edit Distance Dynamic programming . . . . . . Introduction Fibonacci MTP . . Summary . Key components of dynamic programing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • Problems that can be divided into subproblems • Overlapping subproblems - subproblems share subsubproblems • Solves each subsubproblem just once and then saves its answer • Shortest path finding algorithms • DNA sequence alignment • Hidden markov models
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