Rethink the Dijkstra algorithm -- Let's go deeper

Правка en18, от CristianoPenaldo, 2022-10-10 08:53:59

This is a blog for cp newbies (like me).

For a long time I think the Dijkstra algorithm (dij) only has two usages:

(1) Calculate the distance between the vertices when the weights of edges are non-negative.

(2) (Minimax) Given a path $$$p = x_1x_2...x_n$$$, define $$$f(p) := \max_{i=1}^{n-1}d(x_i, x_{i+1})$$$. Given source vertex $$$s$$$ and target vertex $$$t$$$, dij is able to calculate $$$min \{f(p)|p\,\text{is a s-t path}\}$$$.

However, dij works for a function class, not only the sum/max functions. The sum/max functions are only the two most frequently used members of this function class, but the function class is far larger than these two. Once the function $$$f$$$ satisfies several mandatory properties, you could use dij. The word "function class" is like an interface in Go/Java and an abstract class in C++:

~~~~~ struct f{ virtual void property1() = 0; virtual void property2() = 0; virtual void property3() = 0; //... };//dijkstra function. dij(f);

The graph $$$G=(V(G),E(G))$$$. There has to be a non-empty source set S. The S needs not to be a singleton, e.g., multi-source dij. A path p is a vector of vertices {v1,v2,...,vn} and the function f is a function that maps paths to real numbers: path→R. To be brief, f({v1,v2,...,vn}) is shorten to f(v1,v2,...,vn). We say that dij works for f if ∀v∈V(G), dij correctly computes minf({p|pis a S-v path}). f should satisfy three properties that are also sufficient:

(1, induction base) ∀s∈S, f(s) should be correctly initialized.

(2, Extension property) $$$\forall p$$$, for every vertex $$$v$$$ adjacent to the end of $$$p$$$ (p.back()), $$$f(p \cup v) \geq f(p)$$$.

(3, dynamic programming) If path $$$p, q$$$ has the same end (i.e., p.back()==q.back()) and $$$f(p) \geq f(q)$$$, then for every vertex $$$v$$$ adjacent to p.back(), $$$f(p \cup v) \geq f(q \cup v)$$$.

The necessity of the induction base is obvious. Otherwise, the $$$f$$$(source vertices) are wrong. For the second property, it is well known that the sum version of dij (shortest path) does not work for edges with negative cost (but the max version works!). For the third property, let's consider the following example

Теги dijkstra, graph theory, functional programming

История

 
 
 
 
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en77 Английский CristianoPenaldo 2022-10-10 16:15:01 19
en76 Английский CristianoPenaldo 2022-10-10 15:20:42 6
en75 Английский CristianoPenaldo 2022-10-10 15:13:56 2
en74 Английский CristianoPenaldo 2022-10-10 15:09:57 110 (published)
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en72 Английский CristianoPenaldo 2022-10-10 13:37:13 2
en71 Английский CristianoPenaldo 2022-10-10 13:34:11 3
en70 Английский CristianoPenaldo 2022-10-10 13:32:43 0 (published)
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en67 Английский CristianoPenaldo 2022-10-10 13:25:58 127
en66 Английский CristianoPenaldo 2022-10-10 13:24:03 172
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en58 Английский CristianoPenaldo 2022-10-10 12:26:11 43
en57 Английский CristianoPenaldo 2022-10-10 12:24:29 12 Tiny change: 'nature of operator$+$, it satis' -> 'nature of add, it satis'
en56 Английский CristianoPenaldo 2022-10-10 12:24:00 339
en55 Английский CristianoPenaldo 2022-10-10 12:20:57 58
en54 Английский CristianoPenaldo 2022-10-10 12:16:51 2 Tiny change: 'g order:\n- 6, 4, 4, 2' -> 'g order:\n6, 4, 4, 2'
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en46 Английский CristianoPenaldo 2022-10-10 11:35:49 594
en45 Английский CristianoPenaldo 2022-10-10 11:24:49 4 Tiny change: '], k = 5\nOutput: 2\nExplanat' -> '], k = 5\n\nOutput: 2\n\nExplanat'
en44 Английский CristianoPenaldo 2022-10-10 11:24:13 636
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en14 Английский CristianoPenaldo 2022-10-10 08:35:32 142
en13 Английский CristianoPenaldo 2022-10-10 08:19:16 31 Tiny change: 'ion base) For **every** source point $s \in S$, ' -> 'ion base) $\forall s \in S$, '
en12 Английский CristianoPenaldo 2022-10-10 08:18:55 165
en11 Английский CristianoPenaldo 2022-10-10 08:14:54 208
en10 Английский CristianoPenaldo 2022-10-10 08:11:52 514
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en6 Английский CristianoPenaldo 2022-10-10 07:56:08 3 Tiny change: ' \\{f(p)|p \text{is a' -> ' \\{f(p)|p\,\text{is a'
en5 Английский CristianoPenaldo 2022-10-10 07:55:58 14 Tiny change: '\{f(p)|p \in s-t\,paths\\}$\n' -> '\{f(p)|p \text{is a s-t path}\\}$\n'
en4 Английский CristianoPenaldo 2022-10-10 07:55:32 1 Tiny change: 's-t\,paths}\\}$\n' -> 's-t\,paths\\}$\n'
en3 Английский CristianoPenaldo 2022-10-10 07:55:20 16 Tiny change: 'n \\{f(p)|$p$ \text {is a s-t path}\\}$\n' -> 'n \\{f(p)|p \in s-t\,paths}\\}$\n'
en2 Английский CristianoPenaldo 2022-10-10 07:54:52 191 Tiny change: 'p) := \min$\n' -> 'p) := \min_{i=1}^{n-1}d(x_i, x_{i+1})$\n'
en1 Английский CristianoPenaldo 2022-10-10 06:49:27 243 Initial revision (saved to drafts)