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**Greedy Algorithms Technique**

Dr. M. Sakalli, modified from Levitin and CLSR

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Greedy Technique The first key ingredient is the greedy-choice property: a globally optimal solution can be arrived at by making a locally optimal (greedy) choice. In a greedy algorithm, choice is determined on fly at each step, (while algorithm progresses), may seem to be best at the moment and then solves the subproblems after the choice is made. The choice made by a greedy algorithm may be depend on choices so far, but it cannot depend on any future choices or on the solutions to subproblems. Thus, unlike dynamic programming, which solves the subproblems bottom up, a greedy strategy usually progresses in a top-down fashion, making one greedy choice after another, iteratively reducing each given problem instance to a smaller one.

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Algorithms for optimization problems typically go through a sequence of steps, using dynamic programming to determine the best choices, (bottom-up) for subproblem solutions. But in many cases much simpler, more efficient algorithms are possible. A greedy algorithm always makes the choice for a locally optimal solution which seems the best at the current moment with the hope of a globally optimal solution. In the most cases does not always yield optimal solutions, but for many problems. The activity-selection problem, for which a greedy algorithm efficiently computes a solution, works well for a wide range of problems, i.e, minimum-spanning-tree algorithms, Dijkstra's algorithm for shortest paths form a single source, and Chvátal's greedy set-covering heuristic. Greedy algorithms constructs a solution to an optimization problem piece by piece through a sequence of choices that are: feasible locally optimal Irrevocable (binding and abiding)

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**Applications of the Greedy Strategy**

Optimal solutions: change making for “normal” coin denominations minimum spanning tree (MST) single-source shortest paths simple scheduling problems Huffman codes Approximations: traveling salesman problem (TSP) knapsack problem other combinatorial optimization problems

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**Change-Making Problem**

Given unlimited amounts of coins of denominations d1 > … > dm , give change for amount n with the least number of coins Example: d1 = 25c, d2 =10c, d3 = 5c, d4 = 1c and n = 48c Greedy solution: d1+ 2 d2+ 3 d1 Greedy solution is optimal for any amount and “normal’’ set of denominations may not be optimal for arbitrary coin denominations

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Induced subgraph Let a graph be G = {V, E}, we say that H is a subgraph of G and we write HG if the vertices and edges of H = {V’, E’} is a subset of the graph G, V'V, E'E. H need not to accommodate all the edges of G. If the vertices of H are connected and if its all the connecting edges overlay with the edges in G connecting the same adjacent vertices then, H is called induced subgraph. Edge induced, vertex induced, or neither edge nor vertex induced. Note: if at least one path does exist between every pair of vertices, then this is a connected (but not directed yet) graph, directed one is the one whose every edge can only be followed from one vertex to another.

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**Directed acyclic grph, DAG**

Cycle definition seems to be ambiguous, two or three vertices, (only one vertex), if there is a path returning back to the same initial vertex again then there should be a cyclic case too. But!! A cyclic graph is a graph that has at least one cycle (through another vertex -at least one for bi-directional and at least two for uni-directional edged graphs); an acyclic graph is the one that contains no cycles. The girth is the number of the edges involved in the shortest cycle, g>3 is triangle free graph. A source is a vertex with having no incident edges, while a sink is a vertex with no diverging edges from itself. A directed acyclic graph (DAG) is defined from a source to a vertex or from a vertex to a sink, or from a source to a sink, with no directed cycles. A finite DAG must have at least one source and at least one sink. The depth of a vertex in a finite DAG is the length of the longest path from a source to that vertex, while its height is the length of the longest path from that vertex to a sink. The length of a finite DAG is the length (number of edges) of a longest directed path. It is equal to the maximum height of all sources and equal to the maximum depth of all sinks. Wikipedia.

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**Minimum Weight Spanning Tree**

A spanning subgraph is a subgraph that contains all the vertices of the original graph. A spanning tree is a spanning subgraph that is of a acyclic graph of G: (a connected acyclic subgraph) that includes all of vertices of G. Minimum spanning tree of a weighted, connected graph G: a spanning tree of G of minimum total weight. Graph G = {V, E} and the weight function w of a spanning tree T, that connects all V. w(T)=(E, V)ЄT w(u, v) All weights are distinct, “injective” c d b a 6 2 4 3 1 O(E lg V) using binary heaps. O(E + V lg V), using fib heap.

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**Partitioning V of G into A and V-A**

GENERIC-MST(G, w) A Empty set //Initialize while A does not form a spanning tree //Terminate do if edge (u, v) is safe edge for A AA {(u, v)} //Add safe edges return A

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**How to decide to the light edge**

Definitions: a cut (S, V - S) of an undirected graph G = (V, E) is a partition of V, and a light edge is the one, one of its endpoints (vertices) resides in S, and the other in V - S, with minimum possible weight, and a cut respects to A ifthere is no any edge of A crossing the cut. ….… do if edge (u, v) is safe for A In line 3 of the pseudo code given for generic mst(g, w), there must be a spanning tree T such that A T, and if there is an edge of (u, v)T such that (u, v)A, then (u, v) is the one said safe for A. The rule: Theorem

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**How to decide to the light edge**

y x V-S T u S Overlapping subproblems! DP. but MST leads to an efficient algorithm. Theorem: Suppose G = (V, E) is a connected, undirected graph with a real-valued weight function w defined on E, and let A be a subset of E that is included in some MST in G, (S, V - S) be a cut that respects A, and (u, v) be a light edge crossing (S, V - S). Then, edge (u, v) is safe to be united to A. Proof by contradiction: Let T be a mst of G, that AT and suppose another mst T’, with AT’ and a light edge of {u, v}T’ such that (u, v)T. T' (of G) having A{(u, v)} by using a cut-and-paste technique is possible, then (u, v) must be a safe edge for A. But we have another path of T, {x, y} crossing the cut {S, V-S}, (in which one side includes A), This generates a cycle (for which we are paying for two crossings and contradicts to the definition of mst). To minimize the cost we have to exclude the one with the heavier edge, and include the light one. T’ = T - {x, v} {(u, v)} w(u, v) ≤ w(x, y). Therefore, w(T') = w(T) - w(x, y) + w(u, v) ≤w(T) T’ S v

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**Both MST algorithms Kruskal, and Prim, determine a safe edge in line 3 of generic-mst.**

In Kruskal's algorithm, the set A is a forest. - The safe edge added to A is always a least-weight edge connecting two distinct components - many induced subtrees, and gradually merge into each other. In Prim's algorithm, the set A forms a single tree grows like a snowball as one mass. The safe edge added to A is always a least-weighted edge connecting the tree to a vertex not in the tree. Corollary: Let G = (V, E); be a connected, undirected graph, weight function w on E, let A be a subset of E that is included in some MST for G, and C be a connected component (tree) in the forest GA = (V, A). If (u, v) is a light edge connecting C to some other component in GA, then edge (u, v) is safe for A.

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**Kruskal Algorithm It finds a safe edge to add to the growing forest:**

A safe edge (u, v): connecting any two trees in the forest and having the least weight. Let C1 and C2 denote the two trees that are connected by (u, v). Since (u,v) must be a light edge connecting C1 to some other tree, from corollary it must be a safe edge for C1. Then the next step is union of two disjoint trees C1 and C2. Kruskal's algorithm is a greedy algorithm, because at each step it adds to the forest an edge of the least possible weight. Implementation of Kruskal's algorithm employs a disjoint-set data structure to maintain several disjoint sets of elements. Each set contains the vertices in a tree of the current forest. The operation FIND-SET(u) returns a representative element from the set that contains u. Thus, FIND-SET(u) == FIND-SET(v), then vertices u and v belong to the same tree otherwise combine two trees, if {u, v} is a light edge - UNION procedure. As in the other dynamic-set implementations we have studied, each element of a set is represented by an object. Letting x denote an object, we wish to support the following operations. MAKE-SET(x) creates a new set whose only member (and thus representative) is pointed to by x. Since the sets are disjoint, we require that x not already be in a set. UNION(x, y) unites the dynamic sets that contain x and y, say Sx and Sy, into a new set that is the union of these two sets. The two sets are assumed to be disjoint prior to the operation. The representative of the resulting set is some member of Sx Sy, although many implementations of UNION choose the representative of either Sx or Sy, as the new representative. Since we require the sets in the collection to be disjoint, we "destroy" sets Sx and Sy, removing them from the collection S. One of the many applications of disjoint-set data structures arises in determining the connected components of an undirected graph (see Section 5.4). Figure 22.1(a), for example, shows a graph with four connected components. The procedure CONNECTED-COMPONENTS that follows uses the disjoint-set operations to compute the connected components of a graph. Once CONNECTED-COMPONENTS has been run as a preprocessing step, the procedure SAME-COMPONENT answers queries about whether two vertices are in the same connected component.1 (The set of vertices of a graph G is denoted by V[G], and the set of edges is denoted by E[G].) 1When the edges of the graph are "static"--not changing over time--the connected components can be computed faster by using depth-first search (Exercise ). Sometimes, however, the edges are added "dynamically" and we need to maintain the connected components as each edge is added. In this case, the implementation given here can be more efficient than running a new depth-first search for each new edge.

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MST-KRUSKAL(G, w) 1 A Empty set 2 for each v V[G] // for each vertex do MAKE-SET (v) //create |V| trees, 4 sort the edges of E by nondecreasing weight w 5 for each e(u, v) E, in order by nondecreasing weight do if FIND-SET(u) FIND-SET(v) //Check if then A A {(u, v)} //if not in the same set UNION (u, v) //merge two components 9 return A set A to the empty set and The running time for a G = (V, E) depends on the implementation of the disjoint-set data structure. Assume the disjoint-set-forest implementation with the union-by-rank and path-compression heuristics, since it is the asymptotically fastest implementation known. Initialization: O(V), time to sort the edges in line 4: O(E lg E). There are O(E) operations on the disjoint-set forest, which in total take O(E α(E, V)) time, where α is the functional inverse of Ackermann's function defined. Since (E, V) = O(lg E), the total running time of Kruskal's algorithm is O(E lg E).

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Prim’s MST algorithm Operates much like Dijkstra's algorithm for finding shortest paths in a graph. Prim's algorithm has the property that the edges in the set A always form a single tree. Starting from an arbitrary root vertex r (tree A) and expanding one vertex at a time, until the tree spans all the vertices in V. At each turn, a light edge connecting a vertex in A to a vertex in V - A is added to the tree. On each iteration, construct Ti+1 from Ti by adding vertex not in Ti (A) that is closest to those already in Ti (this is a “greedy” step!) The same Corollary: The rule allows merging of the edges that are safe for A; therefore, Terminates when the all vertices are included in A, there the edges in A form a minimum spanning tree. This strategy is "greedy" since the tree is augmented at each step with an edge that contributes the minimum amount possible to the tree's weight. Needs priority queue for locating closest fringe vertex Next analysis rom the notes of CLRS, and listen from MIT

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**Prim Algorithm Example**

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**Notes about Kruskal’s algorithm**

Algorithm looks easier than Prim’s but is harder to implement (checking for cycles!) Cycle checking: a cycle is created iff added edge connects vertices in the same connected component Union-find algorithms

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**Shortest paths – Dijkstra’s algorithm**

Single Source Shortest Paths Problem: Given a weighted connected graph G, find shortest paths from source vertex s to each of the other vertices Dijkstra’s algorithm: Similar to Prim’s MST algorithm, with a different way of computing numerical labels: Among vertices not already in the tree, it finds vertex u with the smallest sum dv + w(v,u) where v is a vertex for which shortest path has been already found on preceding iterations (such vertices form a tree) dv is the length of the shortest path form source to v w(v,u) is the length (weight) of edge from v to u

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**Example Tree vertices Remaining vertices**

b 4 e 3 7 6 2 5 c Example 4 d d Tree vertices Remaining vertices a(-,0) b(a,3) c(-,∞) d(a,7) e(-,∞) a b d 4 c e 3 7 6 2 5 4 b(a,3) c(b,3+4) d(b,3+2) e(-,∞) b c 3 6 2 5 a d e 7 4 d(b,5) c(b,7) e(d,5+4) 4 b c 3 6 2 5 a d 7 e 4 4 c(b,7) e(d,9) b c 3 6 2 5 a d e 7 4 e(d,9)

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**Notes on Dijkstra’s algorithm**

Doesn’t work for graphs with negative weights Applicable to both undirected and directed graphs Efficiency O(|V|2) for graphs represented by weight matrix and array implementation of priority queue O(|E+V|log|V|) for graphs represented by adj. lists and min-heap implementation of priority queue Fib heap O(E+VlogV, amertized.) Don’t mix up Dijkstra’s algorithm with Prim’s algorithm!

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Coding Problem Coding: assignment of bit strings to alphabet characters Codewords: bit strings assigned for characters of alphabet Two types of codes: fixed-length encoding (e.g., ASCII) variable-length encoding (e,g., Morse code) Prefix-free codes: no codeword is a prefix of another codeword Problem: If frequencies of the character occurrences are known, what is the best binary prefix-free code?

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Huffman codes Any binary tree with edges labeled with 0’s and 1’s yields a prefix-free code of characters assigned to its leaves Optimal binary tree minimizing the expected (weighted average) length of a codeword can be constructed as follows Huffman’s algorithm Initialize n one-node trees with alphabet characters and the tree weights with their frequencies. Repeat the following step n-1 times: join two binary trees with smallest weights into one (as left and right subtrees) and make its weight equal the sum of the weights of the two trees. Mark edges leading to left and right subtrees with 0’s and 1’s, respectively.

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**Example character A B C D _ frequency 0.35 0.1 0.2 0.2 0.15**

codeword average bits per character: 2.25 for fixed-length encoding: 3 compression ratio: (3-2.25)/3*100% = 25%

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