19. Dynamic Programming I Memoization, Optimal Substructure, - PowerPoint PPT Presentation

Algorithm RodCut( v , n ) Input: n ≥ 0 , Prices v Output: best value q ← 0 if n > 0 then for i ← 1 , . . . , n do q ← max { q, v i + RodCut ( v, n − i ) } ; return q Running time T ( n ) = � n − 1 ⇒ 39 T ( n ) ∈ Θ(2 n ) i =0 T ( i ) + c 39 T ( n ) = T ( n − 1) + � n − 2 i =0 T ( i ) + c = T ( n − 1) + ( T ( n − 1) − c ) + c = 2 T ( n − 1) ( n > 0) 566

Recursion Tree 5 4 3 2 1 3 2 1 2 1 1 2 1 1 1 1 567

Algorithm RodCutMemoized( m, v, n ) Input: n ≥ 0 , Prices v , Memoization Table m Output: best value q ← 0 if n > 0 then if ∃ m [ n ] then q ← m [ n ] else for i ← 1 , . . . , n do q ← max { q, v i + RodCutMemoized ( m, v, n − i ) } ; m [ n ] ← q return q Running time � n i =1 i = Θ( n 2 ) 568

Subproblem-Graph Describes the mutual dependencies of the subproblems 4 3 2 1 0 and must not contain cycles 569

Construction of the Optimal Cut During the (recursive) computation of the optimal solution for each k ≤ n the recursive algorithm determines the optimal length of the first rod Store the lenght of the first rod in a separate table of length n 570

Bottom-up Description with the example Dimension of the table? Semantics of the entries? 1 571

Bottom-up Description with the example Dimension of the table? Semantics of the entries? 1 n × 1 table. n th entry contains the best value of a rod of length n . 571

Bottom-up Description with the example Dimension of the table? Semantics of the entries? 1 n × 1 table. n th entry contains the best value of a rod of length n . Which entries do not depend on other entries? 2 571

Bottom-up Description with the example Dimension of the table? Semantics of the entries? 1 n × 1 table. n th entry contains the best value of a rod of length n . Which entries do not depend on other entries? 2 Value r 0 is 0 571

Bottom-up Description with the example Dimension of the table? Semantics of the entries? 1 n × 1 table. n th entry contains the best value of a rod of length n . Which entries do not depend on other entries? 2 Value r 0 is 0 What is the execution order such that required entries are always available? 3 . 571

Bottom-up Description with the example Dimension of the table? Semantics of the entries? 1 n × 1 table. n th entry contains the best value of a rod of length n . Which entries do not depend on other entries? 2 Value r 0 is 0 What is the execution order such that required entries are always available? 3 r i , i = 1 , . . . , n . 571

Bottom-up Description with the example Dimension of the table? Semantics of the entries? 1 n × 1 table. n th entry contains the best value of a rod of length n . Which entries do not depend on other entries? 2 Value r 0 is 0 What is the execution order such that required entries are always available? 3 r i , i = 1 , . . . , n . Wie kann sich Lösung aus der Tabelle konstruieren lassen? 4 571

Bottom-up Description with the example Dimension of the table? Semantics of the entries? 1 n × 1 table. n th entry contains the best value of a rod of length n . Which entries do not depend on other entries? 2 Value r 0 is 0 What is the execution order such that required entries are always available? 3 r i , i = 1 , . . . , n . Wie kann sich Lösung aus der Tabelle konstruieren lassen? 4 r n is the best value for the rod of length n . 571

Rabbit! 0 2 0 1 , 1 2 , 1 3 , 1 4 , 1 A rabbit sits on cite (1 , 1) 4 0 3 3 of an n × n grid. It can 1 4 1 1 , 2 2 , 2 3 , 2 4 , 2 only move to east or south. On each pathway there is 3 1 2 4 a number of carrots. How 0 1 1 many carrots does the rab- 1 , 3 2 , 3 3 , 3 4 , 3 bit collect maximally? 0 3 1 3 2 0 2 1 , 4 2 , 4 3 , 4 4 , 4 572

Rabbit! Number of possible paths? Choice of n − 1 ways to south out of 2 n − 2 ways overal. The path 100011 ⇒ No chance for a naive algorithm (1:to south, 0: to east) 573

Rabbit! Number of possible paths? Choice of n − 1 ways to south out of 2 n − 2 ways overal. � 2 n − 2 � ∈ Ω(2 n ) n − 1 The path 100011 ⇒ No chance for a naive algorithm (1:to south, 0: to east) 573

Recursion Wanted: T 0 , 0 = maximal number carrots from (0 , 0) to ( n, n ) . Let w ( i,j ) − ( i ′ ,j ′ ) number of carrots on egde from ( i, j ) to ( i ′ , j ′ ) . Recursion (maximal number of carrots from ( i, j ) to ( n, n )  max { w ( i,j ) − ( i,j +1) + T i,j +1 , w ( i,j ) − ( i +1 ,j ) + T i +1 ,j } , i < n, j < n    w ( i,j ) − ( i,j +1) + T i,j +1 , i = n, j < n  T ij = w ( i,j ) − ( i +1 ,j ) + T i +1 ,j , i < n, j = n    0 i = j = n  574

Graph of Subproblem Dependencies (1 , 1) (2 , 1) (3 , 1) (4 , 1) (1 , 2) (2 , 2) (3 , 2) (4 , 2) (1 , 3) (2 , 3) (3 , 3) (4 , 3) (1 , 4) (2 , 4) (3 , 4) (4 , 4) 575

Bottom-up Description with the example Dimension of the table? Semantics of the entries? 1 576

Bottom-up Description with the example Dimension of the table? Semantics of the entries? 1 Table T with size n × n . Entry at i, j provides the maximal number of carrots from ( i, j ) to ( n, n ) . 576

Bottom-up Description with the example Dimension of the table? Semantics of the entries? 1 Table T with size n × n . Entry at i, j provides the maximal number of carrots from ( i, j ) to ( n, n ) . Which entries do not depend on other entries? 2 576

Bottom-up Description with the example Dimension of the table? Semantics of the entries? 1 Table T with size n × n . Entry at i, j provides the maximal number of carrots from ( i, j ) to ( n, n ) . Which entries do not depend on other entries? 2 Value T n,n is 0 576

Bottom-up Description with the example Dimension of the table? Semantics of the entries? 1 Table T with size n × n . Entry at i, j provides the maximal number of carrots from ( i, j ) to ( n, n ) . Which entries do not depend on other entries? 2 Value T n,n is 0 What is the execution order such that required entries are always available? 3 . 576

Bottom-up Description with the example Dimension of the table? Semantics of the entries? 1 Table T with size n × n . Entry at i, j provides the maximal number of carrots from ( i, j ) to ( n, n ) . Which entries do not depend on other entries? 2 Value T n,n is 0 What is the execution order such that required entries are always available? 3 T i,j with i = n ց 1 and for each i : j = n ց 1 , (or vice-versa: j = n ց 1 and for each j : i = n ց 1 ). 576

Bottom-up Description with the example Dimension of the table? Semantics of the entries? 1 Table T with size n × n . Entry at i, j provides the maximal number of carrots from ( i, j ) to ( n, n ) . Which entries do not depend on other entries? 2 Value T n,n is 0 What is the execution order such that required entries are always available? 3 T i,j with i = n ց 1 and for each i : j = n ց 1 , (or vice-versa: j = n ց 1 and for each j : i = n ց 1 ). Wie kann sich Lösung aus der Tabelle konstruieren lassen? 4 576

Bottom-up Description with the example Dimension of the table? Semantics of the entries? 1 Table T with size n × n . Entry at i, j provides the maximal number of carrots from ( i, j ) to ( n, n ) . Which entries do not depend on other entries? 2 Value T n,n is 0 What is the execution order such that required entries are always available? 3 T i,j with i = n ց 1 and for each i : j = n ց 1 , (or vice-versa: j = n ց 1 and for each j : i = n ց 1 ). Wie kann sich Lösung aus der Tabelle konstruieren lassen? 4 T 1 , 1 provides the maximal number of carrots. 576

Longest Ascending Sequence (LAS) 3 5 6 3 5 6 1 2 4 7 1 2 4 7 3 2 4 6 5 7 1 3 2 4 6 5 7 1 Connect as many as possible fitting ports without lines crossing. 577

Longest Ascending Sequence (LAS) 3 5 6 3 5 6 1 2 4 7 1 2 4 7 3 2 4 6 5 7 1 3 2 4 6 5 7 1 Connect as many as possible fitting ports without lines crossing. 577

Longest Ascending Sequence (LAS) 3 5 6 3 5 6 1 2 4 7 1 2 4 7 3 2 4 6 5 7 1 3 2 4 6 5 7 1 Connect as many as possible fitting ports without lines crossing. 577

Longest Ascending Sequence (LAS) 3 5 6 3 5 6 1 2 4 7 1 2 4 7 3 2 4 6 5 7 1 3 2 4 6 5 7 1 Connect as many as possible fitting ports without lines crossing. 577

Formally Consider Sequence A n = ( a 1 , . . . , a n ) . 1 2 3 4 5 6 7 Search for a longest increasing subsequence of A n . A Examples of increasing subsequences: 3 2 4 6 5 7 1 (3 , 4 , 5) , (2 , 4 , 5 , 7) , (3 , 4 , 5 , 7) , (3 , 7) . 578

Formally Consider Sequence A n = ( a 1 , . . . , a n ) . 1 2 3 4 5 6 7 Search for a longest increasing subsequence of A n . A Examples of increasing subsequences: 3 2 4 6 5 7 1 (3 , 4 , 5) , (2 , 4 , 5 , 7) , (3 , 4 , 5 , 7) , (3 , 7) . Generalization: allow any numbers, even with duplicates (still only strictly increasing subsequences permitted). Example: (2 , 3 , 3 , 3 , 5 , 1) with increasing subsequence (2 , 3 , 5) . 578

First idea Let L i = longest ascending subsequence of A i (1 ≤ i ≤ n ) Assumption: LAS L k of A k known for Now want to compute L k +1 for A k +1 . 579

First idea Let L i = longest ascending subsequence of A i (1 ≤ i ≤ n ) Assumption: LAS L k of A k known for Now want to compute L k +1 for A k +1 . If a k +1 fits to L k , then L k +1 = L k ⊕ a k +1 ? 579

First idea Let L i = longest ascending subsequence of A i (1 ≤ i ≤ n ) Assumption: LAS L k of A k known for Now want to compute L k +1 for A k +1 . If a k +1 fits to L k , then L k +1 = L k ⊕ a k +1 ? Counterexample A 5 = (1 , 2 , 5 , 3 , 4) . Let A 3 = (1 , 2 , 5) with L 3 = A . Determine L 4 from L 3 ? 579

First idea Let L i = longest ascending subsequence of A i (1 ≤ i ≤ n ) Assumption: LAS L k of A k known for Now want to compute L k +1 for A k +1 . If a k +1 fits to L k , then L k +1 = L k ⊕ a k +1 ? Counterexample A 5 = (1 , 2 , 5 , 3 , 4) . Let A 3 = (1 , 2 , 5) with L 3 = A . Determine L 4 from L 3 ? It does not work this way, we cannot infer L k +1 from L k . 579

Second idea. Let L i = longest ascending subsequence of A i (1 ≤ i ≤ n ) Assumption: a LAS L j is known for each j ≤ k . Now compute LAS L k +1 for k + 1 . 580

Second idea. Let L i = longest ascending subsequence of A i (1 ≤ i ≤ n ) Assumption: a LAS L j is known for each j ≤ k . Now compute LAS L k +1 for k + 1 . Look at all fitting L k +1 = L j ⊕ a k +1 ( j ≤ k ) and choose a longest sequence. 580

Second idea. Let L i = longest ascending subsequence of A i (1 ≤ i ≤ n ) Assumption: a LAS L j is known for each j ≤ k . Now compute LAS L k +1 for k + 1 . Look at all fitting L k +1 = L j ⊕ a k +1 ( j ≤ k ) and choose a longest sequence. Counterexample: A 5 = (1 , 2 , 5 , 3 , 4) . Let A 4 = (1 , 2 , 5 , 3) with L 1 = (1) , L 2 = (1 , 2) , L 3 = (1 , 2 , 5) , L 4 = (1 , 2 , 5) . Determine L 5 from L 1 , . . . , L 4 ? 580

Second idea. Let L i = longest ascending subsequence of A i (1 ≤ i ≤ n ) Assumption: a LAS L j is known for each j ≤ k . Now compute LAS L k +1 for k + 1 . Look at all fitting L k +1 = L j ⊕ a k +1 ( j ≤ k ) and choose a longest sequence. Counterexample: A 5 = (1 , 2 , 5 , 3 , 4) . Let A 4 = (1 , 2 , 5 , 3) with L 1 = (1) , L 2 = (1 , 2) , L 3 = (1 , 2 , 5) , L 4 = (1 , 2 , 5) . Determine L 5 from L 1 , . . . , L 4 ? That does not work either: cannot infer L k +1 from only an arbitrary solution L j . We need to consider all LAS. Too many. 580

Third approach Let M n,i = longest ascending subsequence of A i (1 ≤ i ≤ n ) Assumption: the LAS M j for A k , that end with smallest element are known for each of the lengths 1 ≤ j ≤ k . 581

Third approach Let M n,i = longest ascending subsequence of A i (1 ≤ i ≤ n ) Assumption: the LAS M j for A k , that end with smallest element are known for each of the lengths 1 ≤ j ≤ k . Consider all fitting M k,j ⊕ a k +1 ( j ≤ k ) and update the table of the LAS,that end with smallest possible element. 581

Third approach Example Example: A = (1 , 1000 , 1001 , 4 , 5 , 2 , 6 , 7) A LAT M k, · 1 (1) 582

Third approach Example Example: A = (1 , 1000 , 1001 , 4 , 5 , 2 , 6 , 7) A LAT M k, · 1 (1) + 1000 (1) , (1 , 1000) 582

Third approach Example Example: A = (1 , 1000 , 1001 , 4 , 5 , 2 , 6 , 7) A LAT M k, · 1 (1) + 1000 (1) , (1 , 1000) + 1001 (1) , (1 , 1000) , (1 , 1000 , 1001) 582

Third approach Example Example: A = (1 , 1000 , 1001 , 4 , 5 , 2 , 6 , 7) A LAT M k, · 1 (1) + 1000 (1) , (1 , 1000) + 1001 (1) , (1 , 1000) , (1 , 1000 , 1001) + 4 (1) , (1 , 4) , (1 , 1000 , 1001) 582

Third approach Example Example: A = (1 , 1000 , 1001 , 4 , 5 , 2 , 6 , 7) A LAT M k, · 1 (1) + 1000 (1) , (1 , 1000) + 1001 (1) , (1 , 1000) , (1 , 1000 , 1001) + 4 (1) , (1 , 4) , (1 , 1000 , 1001) + 5 (1) , (1 , 4) , (1 , 4 , 5) 582

Third approach Example Example: A = (1 , 1000 , 1001 , 4 , 5 , 2 , 6 , 7) A LAT M k, · 1 (1) + 1000 (1) , (1 , 1000) + 1001 (1) , (1 , 1000) , (1 , 1000 , 1001) + 4 (1) , (1 , 4) , (1 , 1000 , 1001) + 5 (1) , (1 , 4) , (1 , 4 , 5) + 2 (1) , (1 , 2) , (1 , 4 , 5) 582

Third approach Example Example: A = (1 , 1000 , 1001 , 4 , 5 , 2 , 6 , 7) A LAT M k, · 1 (1) + 1000 (1) , (1 , 1000) + 1001 (1) , (1 , 1000) , (1 , 1000 , 1001) + 4 (1) , (1 , 4) , (1 , 1000 , 1001) + 5 (1) , (1 , 4) , (1 , 4 , 5) + 2 (1) , (1 , 2) , (1 , 4 , 5) + 6 (1) , (1 , 2) , (1 , 4 , 5) , (1 , 4 , 5 , 6) 582

Third approach Example Example: A = (1 , 1000 , 1001 , 4 , 5 , 2 , 6 , 7) A LAT M k, · 1 (1) + 1000 (1) , (1 , 1000) + 1001 (1) , (1 , 1000) , (1 , 1000 , 1001) + 4 (1) , (1 , 4) , (1 , 1000 , 1001) + 5 (1) , (1 , 4) , (1 , 4 , 5) + 2 (1) , (1 , 2) , (1 , 4 , 5) + 6 (1) , (1 , 2) , (1 , 4 , 5) , (1 , 4 , 5 , 6) + 7 (1) , (1 , 2) , (1 , 4 , 5) , (1 , 4 , 5 , 6) , (1 , 4 , 5 , 6 , 7) 582

DP Table Idea: save the last element of the increasing sequence M k,j at slot j . 583

DP Table Idea: save the last element of the increasing sequence M k,j 1 2 3 4 5 6 Index at slot j . 3 2 5 1 6 4 Wert Example: 3 2 5 1 6 4 583

DP Table Idea: save the last element of the increasing sequence M k,j 1 2 3 4 5 6 Index at slot j . 3 2 5 1 6 4 Wert Example: 3 2 5 1 6 4 0 1 2 3 4 Index ... - ∞ ∞ ∞ ∞ ∞ ( L j ) j 583

DP Table Idea: save the last element of the increasing sequence M k,j 1 2 3 4 5 6 Index at slot j . 3 2 5 1 6 4 Wert Example: 3 2 5 1 6 4 0 1 2 3 4 Index ... - ∞ ∞ ∞ ∞ ( L j ) j 3 583

DP Table Idea: save the last element of the increasing sequence M k,j 1 2 3 4 5 6 Index at slot j . 3 2 5 1 6 4 Wert Example: 3 2 5 1 6 4 0 1 2 3 4 Index ... - ∞ ∞ ∞ ∞ ( L j ) j 2 583

DP Table Idea: save the last element of the increasing sequence M k,j Index 1 2 3 4 5 6 at slot j . Wert 3 2 5 1 6 4 Example: 3 2 5 1 6 4 Index 0 1 2 3 4 ... ( L j ) j - ∞ 2 5 ∞ ∞ 583

DP Table Idea: save the last element of the increasing sequence M k,j 1 2 3 4 5 6 Index at slot j . 3 2 5 1 6 4 Wert Example: 3 2 5 1 6 4 0 1 2 3 4 Index ... - ∞ ∞ ∞ ( L j ) j 1 5 583

DP Table Idea: save the last element of the increasing sequence M k,j Index 1 2 3 4 5 6 at slot j . Wert 3 2 5 1 6 4 Example: 3 2 5 1 6 4 Index 0 1 2 3 4 ... ( L j ) j - ∞ 1 5 6 ∞ 583

DP Table Idea: save the last element of the increasing sequence M k,j 1 2 3 4 5 6 Index at slot j . 3 2 5 1 6 4 Wert Example: 3 2 5 1 6 4 Problem: Table does not 0 1 2 3 4 Index ... contain the subsequence, only - ∞ ∞ ( L j ) j 1 4 6 the last value. 583

DP Table Idea: save the last element of the increasing sequence M k,j 1 2 3 4 5 6 Index at slot j . 3 2 5 1 6 4 Wert Example: 3 2 5 1 6 4 Problem: Table does not 0 1 2 3 4 Index ... contain the subsequence, only - ∞ ∞ ( L j ) j 1 4 6 the last value. Solution: second table with the predecessors. 583

DP Table Idea: save the last element of the increasing sequence M k,j 1 2 3 4 5 6 Index at slot j . 3 2 5 1 6 4 Wert Example: 3 2 5 1 6 4 2 5 1 Predecessor −∞ −∞ −∞ Problem: Table does not 0 1 2 3 4 Index ... contain the subsequence, only - ∞ ∞ ( L j ) j 1 4 6 the last value. Solution: second table with the predecessors. 583

Dynamic Programming Algorithm LAS Table dimension? Semantics? 1 584

Dynamic Programming Algorithm LAS Table dimension? Semantics? Two tables T [0 , . . . , n ] and V [1 , . . . , n ] . 1 T [ j ] : last Element of the increasing subequence M n,j V [ j ] : Value of the predecessor of a j . Start with T [0] ← −∞ , T [ i ] ← ∞ ∀ i > 1 584

Dynamic Programming Algorithm LAS Table dimension? Semantics? Two tables T [0 , . . . , n ] and V [1 , . . . , n ] . 1 T [ j ] : last Element of the increasing subequence M n,j V [ j ] : Value of the predecessor of a j . Start with T [0] ← −∞ , T [ i ] ← ∞ ∀ i > 1 Computation of an entry 2 584

Dynamic Programming Algorithm LAS Table dimension? Semantics? Two tables T [0 , . . . , n ] and V [1 , . . . , n ] . 1 T [ j ] : last Element of the increasing subequence M n,j V [ j ] : Value of the predecessor of a j . Start with T [0] ← −∞ , T [ i ] ← ∞ ∀ i > 1 Computation of an entry Entries in T sorted in ascending order. For each new entry a k +1 binary 2 search for l , such that T [ l ] < a k < T [ l + 1] . Set T [ l + 1] ← a k +1 . Set V [ k ] = T [ l ] . 584

Dynamic Programming algorithm LAS Computation order 3 585