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Dive deep into Prim's and Kruskal's algorithms for constructing Minimum Spanning Trees efficiently. Follow real-world examples and learn the analysis behind these greedy algorithms step by step.
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PRIM’s ALGORITHM Greedy algorithm. A is the MST being constructed. Thus A is always a Tree. Starts from an arbitrary root. At each step find a light edge crossing the cut (Va, V-Va) where Va = vertices that A is incident on. Add this edge to A. (see image, next slide) How to find the light edge? Idea: Keep V - A in a priority queue, sorted by weight of min-weight edge connecting to A.
1. Array :(unsorted)-> scan to find min, just index and update to change keys. 2. Binary heap: requires reheapify on ExtractMin and DecreaseKey.
KRUSKAL’S ALGORITHM Also a greedy algorithm. Idea: uses the notion of Disjoint Set Union (used for Lab4) Disjoint Set: S is a disjoint set = {S1,S2,...,Sn} st Si Sj = . S is a set of sets, each member set is disjoint. Each set can be viewed as an equivalence class - where all members are equivalent and are represented by one “unique” member.
Operations on a Disjoint Set: • MakeSet(x) S <– S {{x}}; {x} is not in any set Si of S. x is called the representative of the set. • Union(Si, Sj) S<– S – {Si, Sj} {Si Sj}; make new representative from Si and Sj. • FindSet(x) returns Si st x in Si and Si in S. These operations are used in Kruskal’s algorithm.
MSTKruskal(G,W) //uses the Union-Find algorithm T = empty set //T will be MST when done for each v in V do MakeSet(v) //put each vertex in its own equivalence class // make v the class representative sort edges in W by increasing edge weight w for each edge (u,v) in W do //in sorted order if FindSet(u) != FindSet(v) then T = T union {(u,v)} Union( FindSet(u),FindSet(v)) return T
Trace algorithm on example in html notes. • The above disjoint set operations take O(E* a(V,E)) where a is very slow growing (using best known algorithm for disjoint set union–> SS 21.3). • Overall runtime is O(ElgE) which is almost linear if edges are already sorted.
Implementations of a Disjoint Set: Linked List => lots of traversing
Implementations of a Disjoint Set: Augmented Linked List: weight is used to link smaller list into larger one for Union operation, each node points back to head so it is faster to locate the set representative for Find-Set.
Implementations of a Disjoint Set: Forest of Trees:
Implementations of a Disjoint Set: Forest of Trees: Union by Rank, smaller tree joined into larger tree but this can lead to unbalanced trees.
Implementations of a Disjoint Set: Forest of Trees: Balancing the Trees, the Find-Set only needs to traverse one edge in a flat tree.
Implementations of a Disjoint Set: Forest of Trees: Balancing the Trees using Path Compression
Implementations of a Disjoint Set: The punch-line of this discussion is that, taken together, union by rank and path compression produce a spectacularly efficient implementation of the disjoint-set data structure. Theorem 16.2. On a disjoint-set forest with union by rank and path compression, any sequence of m operations, n of which are MAKE-SET operations, has worst-case running time Θ mα(n) , where α is the inverse Ackermann function. Thus, the amortized worst-case running time (see next slide) of each operation is Θ(α(n)). If one makes the approximation α(n) = O(1), which is valid for literally all conceivable purposes, then the operations on a disjoint-set forest have O(1) amortized running time. Because the Ackermann function is an extremely rapidly growing function, the inverse Ackermann function α is an extremely slow growing function (though it is true that lim n→∞ α(n) = ∞).
