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In the algorithm above, steps 1, 2 and 7 will only be run once. For a worst-case evaluation, it should be assumed that step 3 will be run as well. Thus the total amount of time to run steps 1–3 and step 7 is: + + +. The loops in steps 4, 5 and 6 are trickier
Consider finding a shortest path for traveling between two cities by car, as illustrated in Figure 1. Such an example is likely to exhibit optimal substructure. That is, if the shortest route from Seattle to Los Angeles passes through Portland and then Sacramento, then the shortest route from Portland to Los Angeles must pass through Sacramento too.
A planar graph and its minimum spanning tree. Each edge is labeled with its weight, which here is roughly proportional to its length. A minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges of a connected, edge-weighted undirected graph that connects all the vertices together, without any cycles and with the minimum possible total edge weight. [1]
[4] Example (k-colorability Testing Algorithm). Given graph G, choose a random set X of q(ε) = O(k 4 log 2 (k/δ)/ε 3) vertices. For every pair of vertices in X, query if they are adjacent in G. It accepts if the induced subgraph of G on X is k-colorable and rejects otherwise. [4]
Figure 1. Finding the shortest path in a graph using optimal substructure; a straight line indicates a single edge; a wavy line indicates a shortest path between the two vertices it connects (among other paths, not shown, sharing the same two vertices); the bold line is the overall shortest path from start to goal.
The second property implies that a binomial heap with nodes consists of at most + binomial trees, where is the binary logarithm. The number and orders of these trees are uniquely determined by the number of nodes n {\displaystyle n} : there is one binomial tree for each nonzero bit in the binary representation of the number n {\displaystyle ...
The conjecture again remains unproven, but has been resolved for the property of containing a k clique for 2 ≤ k ≤ n. This property is known to have randomized decision tree complexity Θ(n 2). [69] For quantum decision trees, the best known lower bound is Ω(n), but no matching algorithm is known for the case of k ≥ 3. [70]
The potential function method is commonly used to analyze Fibonacci heaps, a form of priority queue in which removing an item takes logarithmic amortized time, and all other operations take constant amortized time. [4] It may also be used to analyze splay trees, a self-adjusting form of binary search tree with logarithmic amortized time per ...