Warshall s Algorithm: Transitive Closure

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1 CS 0 Theory of Algorithms / CS 68 Algorithms in Bioinformaticsi Dynamic Programming Part II. Warshall s Algorithm: Transitive Closure Computes the transitive closure of a relation (Alternatively: all paths in a directed graph) Example of transitive closure: Design and Analysis of Algorithms - Chapter 8 8-

2 Warshall s Algorithm Main idea: a path exists between two vertices i, j, iff there is an edge from i to j; or there is a path from i to j going through vertex ; or there is a path from i to j going through vertex and/or ; or there is a path from i to j going through vertex,, and/or 3; or... there is a path from i to j going through any of the other vertices R R R R R Design and Analysis of Algorithms - Chapter 8 8- Warshall s Algorithm On the k th iteration, the algorithm determine if a path exists between two vertices i, j using just vertices among,,k allowed as intermediate { R (k-) [i,j] (path using just,,k-) R (k) [ij]= [i,j]= or (R (k-) [i,k] and R (k-) [k,j]) (path from i to k k and from k to i using just,,k-) i k th iteration j Design and Analysis of Algorithms - Chapter 8 8-3

3 Warshall s Algorithm: Transitive Closure Design and Analysis of Algorithms - Chapter 8 8- Warshall s Algorithm (matrix generation) Recurrence relating elements R (k) to elements of R (k-) is: R (k) [i,j] = R (k-) [i,j] or (R (k-) [i,k] and R (k-) [k,j]) It implies the following rules for generating R (k) from R (k-) : Rule If an element in row i and column j is in R (k-), it remains in R (k) Rule If an element in row i and column j is 0 in R (k-), it has to be changed to in R (k) if and only if the element in its row i and column k and the element in its column j and row k are both s inr R (k-)

4 Warshall s Algorithm: Transitive Closure Design and Analysis of Algorithms - Chapter Design and Analysis of Algorithms - Chapter 8 8-7

5 Warshall s Algorithm (pseudocode and analysis) Time efficiency: (n 3 ) Space efficiency: Matrices can be written over their predecessors Floyd s Algorithm: All pairs shortest paths Problem: In a weighted (di)graph, find shortest paths between every pair of vertices Same idea: construct solution through series of matrices D (0),, D (n) using increasing subsets of the vertices allowed as intermediate Example: Design and Analysis of Algorithms - Chapter 8 8-9

6 Floyd s Algorithm (matrix generation) On the k-th iteration, the algorithm determines shortest paths between every pair of vertices i, j that use only vertices among,,k as intermediate D (k) [i,j] = min {D (k-) [i,j], D (k-) [i,k] + D (k-) [k,j]} D (k-) [i,k] i D (k-) [i,j] j k D (k-) [k,j] Floyd s Algorithm: All pairs shortest paths Design and Analysis of Algorithms - Chapter 8 8-

7 Design and Analysis of Algorithms - Chapter 8 8- Floyd s Algorithm (pseudocode and analysis) Timeefficiency: ce cy: ( (n 3 ) Space efficiency: Matrices can be written over their predecessors

8 Knapsack Problem by DP Given n items of integer weights: w w w n values: v v v n a knapsack of integer capacity W find most valuable subset of the items that fit into the knapsack Consider instance defined by first i items and capacity j (j W). Let V[i,j] be optimal value of such instance. Then V[i,j]= max {V[i-,j], v i + V[i-,j- w i ]} if j- w i 0 V[i-,j] if j- w i < 0 Initial conditions: V[0,j] = 0 and V[i,0] = 0 Knapsack Problem by DP (example) Example: Knapsack of capacity W = 5 item weight value $ $0 3 3 $0 $5 capacity yjj w =, v = w =, v = 0 w 3 = 3, v 3 = 0 3 w =, v = 5?

9 Knapsack Problem V[i, j] = max (V[i, j], V[i, j w i ] + v i ) object i not used object i used Design and Analysis of Algorithms - Chapter Knapsack Problem V[i, j] = max (V[i, j], V[i, j w i ] + v i ) object i not used object i used Design and Analysis of Algorithms - Chapter 8 8-7

10 Knapsack Problem Memory Function Implement the recurrence recursively Do not calculate a value if it is not needed Do not recalculate a value Row 0 and column 0 of V are initialized to 0, other entries are - MFKnapsack(i, j) if V[i, j]< 0 if j < w[i] value MFKnapsack(i, j) else value max (MFKnapsack(i, j), MFKnapsack(i,j w[i]) + v[i]) V[i, j] value return V[i, j] Design and Analysis of Algorithms - Chapter Knapsack Problem Memory Function Design and Analysis of Algorithms - Chapter 8 8-9

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