Movatterモバイル変換

Design and Analysis of AlgorithmGreedy Methods(Single Source shortest path,Knapsack problem )Lecture – 45 - 53

• A greedy algorithm always makes the choice thatlooks best at the moment. (i.e. it makes a locallyoptimal choice in the hope that this choice willlead to a globally optimal solution).• The objective of this section is to exploresoptimization problems that are solvable by greedyalgorithms.Overview

Greedy Algorithm• In mathematics, computer science andeconomics, an optimization problem is theproblem of finding the best solution from allfeasible solutions.• Algorithms for optimization problems typically gothrough a sequence of steps, with a set ofchoices at each step.• Many optimization problems can be solved usinga greedy approach.• Greedy algorithms are simple andstraightforward.

Greedy Algorithm• A greedy algorithm always makes the choice thatlooks best at the moment.• That is, it makes a locally optimal choice in thehope that this choice will lead to a globallyoptimal solution.• Greedy algorithms do not always yield optimalsolutions, but for many problems they do.• This algorithms are easy to invent, easy toimplement and most of the time provides bestand optimized solution.

Greedy Algorithm• Application of Greedy Algorithm:• A simple but nontrivial problem, the activity-selection problem, for which a greedy algorithmefficiently computes a solution.• In combinatorics,(a branch of mathematics), a‘matroid’ is a structure that abstracts andgeneralizes the notion of linear independence invector spaces. Greedy algorithm always producesan optimal solution for such problems. Schedulingunit-time tasks with deadlines and penalties is anexample of such problem.

Greedy Algorithm• Application of Greedy Algorithm:• An important application of greedy techniques isthe design of data-compression codes (i.e.Huffman code) .• The greedy method is quite powerful and workswell for a wide range of problems. They are:• Minimum-spanning-tree algorithms(Example: Prims and Kruskal algorithm)• Single Source Shortest Path.(Example: Dijkstra's and Bellman ford algorithm)

Greedy Algorithm• Application of Greedy Algorithm:• A problem exhibits optimal substructure if anoptimal solution to the problem contains within itoptimal solutions to subproblems.• This property is a key ingredient of assessing theapplicability of dynamic programming as well asgreedy algorithms.• The subtleties between the above two techniquesare illustrated with the help of two variants of aclassical optimization problem known as knapsackproblem. These variants are:• 0-1 knapsack problem (Dynamic Programming)• Fractional knapsack problem (Greedy Algorithm)

Greedy Algorithm• Problem 5: Single source shortest path• It is a shortest path problem where the shortestpath from a given source vertex to all otherremaining vertices is computed.• Dijkstra’s Algorithm and Bellman FordAlgorithm are the famous algorithms used forsolving single-source shortest path problem.

Greedy Algorithm• Problem 5: Single source shortest path• Dijkstra’s Algorithm• Dijkstra Algorithm is a very famous greedyalgorithm.• It is used for solving the single sourceshortest path problem.• It computes the shortest path from oneparticular source node to all other remainingnodes of the graph.

Greedy Algorithm• Problem 5: Single source shortest path• Dijkstra’s Algorithm (Feasible Condition)• Dijkstra algorithm works• for connected graphs.• for those graphs that do not contain anynegative weight edge.• To provides the value or cost of theshortest paths.• for directed as well as undirected graphs.

Greedy Algorithm• Problem 5: Single source shortest path• Dijkstra’s Algorithm (Implementation)The implementation of Dijkstra Algorithm is executed in thefollowing steps-• Step-01:• In the first step. two sets are defined-• One set contains all those vertices which have been includedin the shortest path tree.• In the beginning, this set is empty.• Other set contains all those vertices which are still left to beincluded in the shortest path tree.• In the beginning, this set contains all the vertices of thegiven graph.

Greedy Algorithm• Problem 5: Single source shortest path• Dijkstra’s Algorithm (Implementation)The implementation of Dijkstra Algorithm is executed in thefollowing steps-• Step-02:For each vertex of the given graph, two variables are definedas-• Π[v] which denotes the predecessor of vertex ‘v’• d[v] which denotes the shortest path estimate of vertex ‘v’from the source vertex.

Greedy Algorithm• Problem 5: Single source shortest path• Dijkstra’s Algorithm (Implementation)The implementation of Dijkstra Algorithm is executed in thefollowing steps-• Step-02:Initially, the value of these variables is set as-• The value of variable ‘Π’ for each vertex is set to NIL i.e.Π[v] = NIL• The value of variable ‘d’ for source vertex is set to 0 i.e. d[S]= 0• The value of variable ‘d’ for remaining vertices is set to ∞i.e. d[v] = ∞

Greedy Algorithm• Problem 5: Single source shortest path• Dijkstra’s Algorithm (Implementation)The implementation of Dijkstra Algorithm is executed in thefollowing steps-• Step-03:The following procedure is repeated until all the vertices of thegraph are processed-• Among unprocessed vertices, a vertex with minimum valueof variable ‘d’ is chosen.• Its outgoing edges are relaxed.• After relaxing the edges for that vertex, the sets created instep-01 are updated.

Greedy Algorithm• Problem 5: Single source shortest path• Dijkstra’s AlgorithmExample 1: Construct the Single source shortest path for thegiven graph using Dijkstra’s Algorithm-4∞s∞x∞t∞y∞z2 971623510

Greedy Algorithm• Problem 5: Single source shortest path• Dijkstra’s AlgorithmExample 1: Construct the Single source shortest path for thegiven graph using Dijkstra’s Algorithm-40s∞x∞t∞y∞z2 971623510

Greedy Algorithm• Problem 5: Single source shortest path• Dijkstra’s AlgorithmExample 1: Construct the Single source shortest path for thegiven graph using Dijkstra’s Algorithm-40s∞x10t5y∞z2 971623510

Greedy Algorithm• Problem 5: Single source shortest path• Dijkstra’s AlgorithmExample 1: Construct the Single source shortest path for thegiven graph using Dijkstra’s Algorithm-40s14x8t5y7z2 971623510

Greedy Algorithm• Problem 5: Single source shortest path• Dijkstra’s AlgorithmExample 1: Construct the Single source shortest path for thegiven graph using Dijkstra’s Algorithm-40s9x8t5y7z2 971623510Hence the shortest path toall the vertex from s are:𝑠 → 𝑡 = 8𝑠 → 𝑥 = 9𝑠 → 𝑦 = 5𝑠 → 𝑧 = 7

Greedy Algorithm• Problem 5: Single source shortest path• Dijkstra’s Algorithm (Algorithm)DIJKSTRA(G, w, s)1 INITIALIZE-SINGLE-SOURCE(G, s)2 S ← Ø3 Q ← V[G]4 while Q ≠ Ø5 do u ← EXTRACT-MIN(Q)6 S ← 𝑆 ∈ {𝑢}7 for each vertex 𝑣 ∈ 𝐴𝑑𝑗[𝑢]8 do RELAX(u, v, w)

Greedy Algorithm• Problem 5: Single source shortest path• Dijkstra’s Algorithm (Algorithm)INITIALIZE-SINGLE-SOURCE(G, s)1 for each vertex v V[G]2 do d[v] ← ∞3 π[v] ← NIL4 d[s] ← 0RELAX(u, v, w)1 if d[v] > d[u] + w(u, v)2 then d[v] ← d[u] + w(u, v)3 π[v] ← u

Greedy Algorithm• Problem 5: Single source shortest path• Dijkstra’s Algorithm (Complexity)CASE-01:• 𝐴[𝑖, 𝑗] stores the information about edge (𝑖, 𝑗).• Time taken for selecting 𝑖 with the smallest 𝑑𝑖𝑠𝑡 is 𝑂(𝑉).• For each neighbor of i, time taken for updating 𝑑𝑖𝑠𝑡[𝑗] is 𝑂(1)and there will be maximum V-1 neighbors.• Time taken for each iteration of the loop is O(V) and onevertex is deleted from Q.• Thus, total time complexity becomes O(𝑉2).

Greedy Algorithm• Problem 5: Single source shortest path• Dijkstra’s Algorithm (Complexity)CASE-02:• With adjacency list representation, all vertices of the graphcan be traversed using BFS in 𝑂(𝑉 + 𝐸) time.• In min heap, operations like extract-min and decrease-keyvalue takes 𝑂(log 𝑉) time.• So, overall time complexity becomes𝑂(𝐸 + 𝑉) 𝑥 𝑂(log 𝑉) which is 𝑂((𝐸 + 𝑉) 𝑥 log 𝑉) = 𝑂(𝐸 log 𝑉)

Greedy Algorithm• Problem 5: Single source shortest path• Dijkstra’s Algorithm (Self Practice)Example 2: Construct the Single source shortest path for thegiven graph using Dijkstra’s Algorithm-s∞c∞a∞d∞b∞e∞11125222

Greedy Algorithm• Problem 5: Single source shortest path• Bellman Ford Algorithm• Bellman-Ford algorithm solves the single-sourceshortest-path problem in the general case in whichedges of a given digraph can have negative weightas long as G contains no negative cycles.• Like Dijkstra's algorithm, this algorithm, uses thenotion of edge relaxation but does not use withgreedy method. Again, it uses d[u] as an upperbound on the distance d[u, v] from u to v.

Greedy Algorithm• Problem 5: Single source shortest path• Bellman Ford Algorithm• The algorithm progressively decreases an estimated[v] on the weight of the shortest path from thesource vertex s to each vertex v ∈ V until it achievethe actual shortest-path.• The algorithm returns Boolean TRUE if the givendigraph contains no negative cycles that arereachable from source vertex s otherwise it returnsBoolean FALSE.

Greedy Algorithm• Problem 5: Single source shortest pathBellman Ford Algorithm (Negative CycleDetection)Assume:𝑑[𝑢] ≤ 𝑑[𝑥] + 4𝑑[𝑣] ≤ 𝑑[𝑢] + 5𝑑[𝑥] ≤ 𝑑[𝑣] − 10Adding:𝑑[𝑢] + 𝑑[𝑣] + 𝑑[𝑥] ≤ 𝑑[𝑥] + 𝑑[𝑢] + 𝑑[𝑣] − 1Because it’s a cycle, vertices on left are same as those on right. Thuswe get 0 ≤ −1; a contradiction.So for at least one edge (𝑢, 𝑣),𝑑[𝑣] > 𝑑[𝑢] + 𝑤(𝑢, 𝑣)This is exactly what Bellman-Ford checks for.uxv45-10

Greedy Algorithm• Problem 5: Single source shortest path• Bellman Ford Algorithm (Implementation)• Step 1: Start with the weighted graph.• Step 2: Choose the starting vertex by making the pathvalue zero and assign infinity path values to all othervertices.• Step 3: Visit each edge and relax the path distances ifthey are inaccurate.• Step 4: Do step 3 V-1 times because in the worst case avertex's path length might need to be readjusted V-1times.• Step 5: After all vertices have their path lengths, check ifa negative cycle is present or not.

Greedy Algorithm• Problem 5: Single source shortest path• Bellman Ford AlgorithmExample 1: Construct the Single source shortest path for thegiven graph using Bellman ford Algorithm-∞s∞x∞t∞y∞z8-32579-276-4

Greedy Algorithm• Problem 5: Single source shortest path• Bellman Ford AlgorithmExample 1: Construct the Single source shortest path for thegiven graph using Bellman ford Algorithm-∞s∞x∞t∞y∞z8-32579-276-40s∞x∞t∞y∞z8-32579-276-4Iteration - 1

Greedy Algorithm• Problem 5: Single source shortest path• Bellman Ford AlgorithmExample 1: Construct the Single source shortest path for thegiven graph using Bellman ford Algorithm-0s∞x6t∞y∞z8-32579-276-4Iteration - 1 Edge no - 10s∞x∞t∞y∞z8-32579-276-4

Greedy Algorithm• Problem 5: Single source shortest path• Bellman Ford AlgorithmExample 1: Construct the Single source shortest path for thegiven graph using Bellman ford Algorithm-0s∞x6t7y∞z8-32579-276-4Iteration - 1 Edge no - 20s∞x∞t∞y∞z8-32579-276-4

Greedy Algorithm• Problem 5: Single source shortest path• Bellman Ford AlgorithmExample 1: Construct the Single source shortest path for thegiven graph using Bellman ford Algorithm-0s∞x6t7y∞z8-32579-276-4Iteration - 1 Edge no – 100s∞x∞t∞y∞z8-32579-276-4

Greedy Algorithm• Problem 5: Single source shortest path• Bellman Ford AlgorithmExample 1: Construct the Single source shortest path for thegiven graph using Bellman ford Algorithm-0s∞x6t∞y∞z8-32579-276-4Iteration - 2 Edge no - 10s∞x6t7y∞z8-32579-276-4

Greedy Algorithm• Problem 5: Single source shortest path• Bellman Ford AlgorithmExample 1: Construct the Single source shortest path for thegiven graph using Bellman ford Algorithm-0s∞x6t7y∞z8-32579-276-4Iteration - 2 Edge no - 20s∞x6t7y∞z8-32579-276-4

Greedy Algorithm• Problem 5: Single source shortest path• Bellman Ford AlgorithmExample 1: Construct the Single source shortest path for thegiven graph using Bellman ford Algorithm-0s∞x6t7y16z8-32579-276-4Iteration - 2 Edge no - 30s∞x6t7y∞z8-32579-276-4

Greedy Algorithm• Problem 5: Single source shortest path• Bellman Ford AlgorithmExample 1: Construct the Single source shortest path for thegiven graph using Bellman ford Algorithm-0s11x6t7y16z8-32579-276-4Iteration - 2 Edge no - 60s∞x6t7y∞z8-32579-276-4

Greedy Algorithm• Problem 5: Single source shortest path• Bellman Ford AlgorithmExample 1: Construct the Single source shortest path for thegiven graph using Bellman ford Algorithm-0s4x6t7y2z8-32579-276-4Iteration - 2 Edge no – 100s∞x6t7y∞z8-32579-276-4

Greedy Algorithm• Problem 5: Single source shortest path• Bellman Ford AlgorithmExample 1: Construct the Single source shortest path for thegiven graph using Bellman ford Algorithm-0s4x6t7y2z8-32579-276-4Iteration - 3 Edge no - 10s4x6t7y2z8-32579-276-4

Greedy Algorithm• Problem 5: Single source shortest path• Bellman Ford AlgorithmExample 1: Construct the Single source shortest path for thegiven graph using Bellman ford Algorithm-Iteration - 3 Edge no - 20s4x6t7y2z8-32579-276-40s4x6t7y2z8-32579-276-4

Greedy Algorithm• Problem 5: Single source shortest path• Bellman Ford AlgorithmExample 1: Construct the Single source shortest path for thegiven graph using Bellman ford Algorithm-Iteration - 3 Edge no – 100s4x6t7y2z8-32579-276-40s4x2t7y2z8-32579-276-4

Greedy Algorithm• Problem 5: Single source shortest path• Bellman Ford AlgorithmExample 1: Construct the Single source shortest path for thegiven graph using Bellman ford Algorithm-Iteration - 4 Edge no - 10s4x2t7y2z8-32579-276-4 0s4x2t7y2z8-32579-276-4

Greedy Algorithm• Problem 5: Single source shortest path• Bellman Ford AlgorithmExample 1: Construct the Single source shortest path for thegiven graph using Bellman ford Algorithm-Iteration - 4 Edge no - 70s4x2t7y−2z8-32579-276-40s4x2t7y2z8-32579-276-4

Greedy Algorithm• Problem 5: Single source shortest path• Bellman Ford AlgorithmExample 1: Construct the Single source shortest path for thegiven graph using Bellman ford Algorithm-Iteration - 4 Edge no - 80s4x2t7y2z8-32579-276-4 0s4x2t7y−2z8-32579-276-4

Greedy Algorithm• Problem 5: Single source shortest path• Bellman Ford AlgorithmExample 1: Construct the Single source shortest path for thegiven graph using Bellman ford Algorithm-Iteration - 4 Edge no - 90s4x2t7y−2z8-32579-276-40s4x2t7y2z8-32579-276-4

Greedy Algorithm• Problem 5: Single source shortest path• Bellman Ford AlgorithmExample 1: Construct the Single source shortest path for thegiven graph using Bellman ford Algorithm-Iteration - 4 Edge no – 100s4x2t7y2z8-32579-276-4 0s4x2t7y−2z8-32579-276-4

Greedy Algorithm• Problem 5: Single source shortest path• Bellman Ford AlgorithmExample 1: Construct the Single source shortest path for the given graphusing Bellman ford Algorithm-0s4x2t7y−2z8-32579-276-40s∞x∞t∞y∞z8-32579-276-40s∞x6t7y∞z8-32579-276-40s4x6t7y2z8-32579-276-40s4x2t7y2z8-32579-276-4After Iteration 1After Iteration 3After Iteration 2After Iteration 4

Greedy Algorithm• Problem 5: Single source shortest path• Bellman Ford Algorithm (Algorithm)BELLMAN-FORD(G, w, s)1 INITIALIZE-SINGLE-SOURCE(G, s)2 for i ← 1 to |V[G]| - 13 do for each edge (u, v) ∈ E[G]4 do RELAX(u, v, w)5 for each edge (u, v) ∈ E[G]6 do if d[v] > d[u] + w(u, v)7 then return FALSE8 return TRUE

Greedy Algorithm• Problem 5: Single source shortest path• Bellman Ford Algorithm (Algorithm)INITIALIZE-SINGLE-SOURCE(G, s)1 for each vertex v V[G]2 do d[v] ← ∞3 π[v] ← NIL4 d[s] ← 0RELAX(u, v, w)1 if d[v] > d[u] + w(u, v)2 then d[v] ← d[u] + w(u, v)3 π[v] ← u

Greedy Algorithm• Problem 5: Single source shortest path• Bellman Ford Algorithm (Analysis)BELLMAN-FORD(G, w, s)1 INITIALIZE-SINGLE-SOURCE(G, s) → Θ(𝑉)2 for i ← 1 to |V[G]| - 13 do for each edge (u, v) ∈ E[G] Ο(𝐸)4 do RELAX(u, v, w)5 for each edge (u, v) ∈ E[G]6 do if d[v] > d[u] + w(u, v) Ο(𝐸)7 then return FALSE8 return TRUEΟ(𝐸)

Greedy Algorithm• Problem 5: Single source shortest path• Bellman Ford Algorithm(Self Practice)Example 2: Construct the Single source shortest path for thegiven graph using Bellman Ford Algorithm-s∞c∞a∞d∞b∞e∞11125222

Greedy Algorithm• Problem 5: Knapsack ProblemProblem definition:• The Knapsack problem is an “combinatorialoptimization problem”, a topic in mathematicsand computer science about finding the optimalobject among a set of object .• Given a set of items, each with a weight and aprofit, determine the number of each item toinclude in a collection so that the total weight isless than or equal to a given limit and the totalprofit is as large as possible.

Greedy Algorithm• Problem 5: Knapsack ProblemVersions of Knapsack:• Fractional Knapsack Problem:Items are divisible; you can take any fraction ofan item and it is solved by using GreedyAlgorithm.• 0/1 Knapsack Problem:Items are indivisible; you either take them ornot and it is solved by using DynamicProgramming(DP).

Greedy Algorithm• Problem 5: Knapsack Problem• Fractional Knapsack Problem:Given n objects and a knapsack with a capacity“M” (weight)• Each object 𝑖 has weight 𝑤𝑖 and profit 𝑝𝑖.• For each object 𝑖, suppose a fraction 𝑥𝑖, 0 ≤ 𝑥𝑖 ≤ 1, can be placed in theknapsack, then the profit earned is 𝑝𝑖. 𝑥𝑖 .

Greedy Algorithm• Problem 5: Knapsack ProblemFractional Knapsack Problem:The objective is to maximize profit subject to capacity constraints.𝑖. 𝑒. 𝑚𝑎𝑥𝑖𝑚𝑖𝑧𝑒𝑖=1𝑛𝑝𝑖𝑥𝑖 ------------------------------(1)Subject to𝑖=1𝑛𝑤𝑖𝑥𝑖 ≤ 𝑀 ------------------------(2)Where, 0 ≤ 𝑥𝑖 ≤ 1,𝑝𝑖 > 0,𝑤𝑖 > 0.A feasible solution is any subset {𝑥1, 𝑥2, 𝑥3, … … , 𝑥𝑛} satisfying (1) & (2).An optimal solution is a feasible solution that maximize 𝑖=1𝑛𝑝𝑖𝑥𝑖.

Greedy Algorithm• Problem 5: Knapsack ProblemFractional Knapsack Problem(Implementation)Fractional knapsack problem is solved using greedy method in thefollowing steps-Step-01:• For each item, compute its (profit / weight) ratio.(i.e 𝑝𝑖/𝑥𝑖)Step-02:• Arrange all the items in decreasing order of their (profit /weight) ratio.Step-03:• Start putting the items into the knapsack beginning from theitem with the highest ratio.• Put as many items as you can into the knapsack.

Greedy Algorithm• Problem 5: Knapsack ProblemFractional Knapsack Problem(Implementation)Example 1: For the given set of items and knapsack capacity = 60 kg,find the optimal solution for the fractional knapsack problem making useof greedy approach.Step-01: Compute the (profit / weight) ratio for each item-

Greedy Algorithm• Problem 5: Knapsack ProblemFractional Knapsack Problem(Algorithm)FRACTIONAL_KNAPSACK (p, 𝑤, 𝑀)1 for i = 1 to n2 do x[i] = 03 weight = 04 for i = 1 to n5 if weight + w[i] ≤ M6 then x[i] = 17 weight = weight + w[i]8 else9 x[i] = (M - weight) / w[i]10 weight = M11 break12 return x

Greedy Algorithm• Problem 5: Knapsack ProblemFractional Knapsack Problem(Implementation)Example 1: For the given set of items and knapsack capacity = 60 kg,find the optimal solution for the fractional knapsack problem making useof greedy approach.Item Weight Profit1 5 302 10 403 15 454 22 775 25 90

Greedy Algorithm• Problem 5: Knapsack ProblemFractional Knapsack Problem(Implementation)Example 1: For the given set of items and knapsack capacity = 60 kg,find the optimal solution for the fractional knapsack problem making useof greedy approach.Step-01: Compute the (profit / weight) ratio for each item-Item Weight Profit Ratio1 5 30 62 10 40 43 15 45 34 22 77 3.55 25 90 3.6

Greedy Algorithm• Problem 5: Knapsack ProblemFractional Knapsack Problem(Implementation)Example 1: For the given set of items and knapsack capacity = 60 kg,find the optimal solution for the fractional knapsack problem making useof greedy approach.Step-02: Sort all the items in decreasing order of their value / weightratio-Item Weight Profit Ratio1 5 30 62 10 40 45 25 90 3.64 22 77 3.53 15 45 3

Greedy Algorithm• Problem 5: Knapsack ProblemFractional Knapsack Problem(Implementation)Example 1: For the given set of items and knapsack capacity = 60 kg,find the optimal solution for the fractional knapsack problem making useof greedy approach.Step-03: Start filling the knapsack by putting the items into it one byone.Item Weight Profit Ratio1 5 30 62 10 40 45 25 90 3.64 22 77 3.53 15 45 3KnapsackweightItems inKnapsack Cost60 ∅ 0

Greedy Algorithm• Problem 5: Knapsack ProblemFractional Knapsack Problem(Implementation)Example 1: For the given set of items and knapsack capacity = 60 kg,find the optimal solution for the fractional knapsack problem making useof greedy approach.Step-03: Start filling the knapsack by putting the items into it one byone.Item Weight Profit Ratio1 5 30 62 10 40 45 25 90 3.64 22 77 3.53 15 45 3KnapsackweightItems inKnapsack Cost60 ∅ 055 1 30

Greedy Algorithm• Problem 5: Knapsack ProblemFractional Knapsack Problem(Implementation)Example 1: For the given set of items and knapsack capacity = 60 kg,find the optimal solution for the fractional knapsack problem making useof greedy approach.Step-03: Start filling the knapsack by putting the items into it one byone.Item Weight Profit Ratio1 5 30 62 10 40 45 25 90 3.64 22 77 3.53 15 45 3KnapsackweightItems inKnapsack Cost60 ∅ 055 1 3045 1, 2 70

Greedy Algorithm• Problem 5: Knapsack ProblemFractional Knapsack Problem(Implementation)Example 1: For the given set of items and knapsack capacity = 60 kg,find the optimal solution for the fractional knapsack problem making useof greedy approach.Step-03: Start filling the knapsack by putting the items into it one byone.Item Weight Profit Ratio1 5 30 62 10 40 45 25 90 3.64 22 77 3.53 15 45 3KnapsackweightItems inKnapsack Cost60 ∅ 055 1 3045 1,2 7020 1,2,5 160

Greedy Algorithm• Problem 5: Knapsack ProblemFractional Knapsack Problem(Implementation)Example 1:Now, Knapsack weight left to be filled is 20 kg but item-4 has a weight of 22 kg.Since in fractional knapsack problem, even the fraction of any item can be taken.So, knapsack will contain the fractional part of item 4.(20 out of 22)Total cost of the knapsack = 160 + (20/22) x 77 = 160 + 70 = 230 unitsItem Weight Profit Ratio1 5 30 62 10 40 45 25 90 3.64 22 77 3.53 15 45 3KnapsackweightItems inKnapsack Cost60 ∅ 055 1 3045 1,2 7020 1,2,5 16001,2,5,frac(4) 230

Greedy Algorithm• Problem 5: Knapsack ProblemFractional Knapsack Problem(Algorithm)• The main time taking step is the sorting of all items indecreasing order of their (profit / weight) ratio.• If the items are already arranged in the required order,then while loop takes 𝑂(𝑛) time.• The average time complexity of Quick Sort is 𝑂(𝑛 log 𝑛).• Therefore, total time taken including the sort is 𝑂(𝑛 log 𝑛).

Greedy Algorithm• Problem 5: Knapsack ProblemFractional Knapsack Problem(self practice)Example 2: For the given set of items and knapsack capacity = 60 kg,find the optimal solution for the fractional knapsack problem making useof greedy approach.Item Weight ProfitA 1 5B 3 9C 2 4D 2 8

Greedy Algorithm• Problem 5: Knapsack ProblemFractional Knapsack Problem(self practice)Example 3: Find the optimal solution for the fractional knapsackproblem making use of greedy approach. Consider-n = 3M = 20 kg(w1, w2, w3) = (18, 15, 10)(p1, p2, p3) = (25, 24, 15)

Single source Shortest path algorithm with example

Movatterモバイル変換

Change Language

Single source Shortest path algorithm with example

Embed presentation

Recommended

More Related Content

What's hot

Similar to Single source Shortest path algorithm with example

Recently uploaded

In this document

Single source Shortest path algorithm with example