TL;DR

• Dijkstra’s algorithm is the classic way to compute shortest paths with non-negative weights, running in about O(m + n log n).
• A 2025 result gives a deterministic SSSP for directed graphs that runs in O(m · (log n)^(2/3)) in the comparison-addition model. It beats the long-standing n log n “sorting barrier” on sparse graphs.
• Big idea in plain words: instead of strictly sorting by distance at every step like Dijkstra, the new method processes nodes in smart batches using bounded multi-source relaxations and a reduced frontier of pivots.

The problem in one minute

Single-Source Shortest Path (SSSP) asks: given a source node, what is the minimum distance to every other node in a weighted graph. Classic answer: Dijkstra’s algorithm for non-negative weights. It repeatedly picks the closest unfinished node, then relaxes its outgoing edges. That “pick the closest” step creates a sorting-like cost of about log n per operation, which is why the total time has the n log n behavior.

Researchers showed a new deterministic way that reduces the dependency on sorting by exploring nodes in layers and shrinking the active frontier so heavy work applies only to a small subset. This changes the asymptotic behavior to O(m · (log n)^(2/3)).

Visuals to anchor the intuition

1) Growth trend: n·log n vs n·(log n)^(2/3)

What you see: both grow with n, but n·(log n)^(2/3) grows a bit slower than n·log n. For huge graphs, that gap matters.

2) What changes operationally

Left: Dijkstra picks exactly the closest next node each step. Right: the new mindset processes a batch frontier and runs a few k-hop relaxations, then moves the frontier forward.

3) Tiny graph we will use in code

Edges: A→B(2), A→C(2), B→C(1), B→D(3), C→D(2), C→E(3), D→E(1) Expected distances from A: A 0, B 2, C 2, D 4, E 5

Practical example 1: Dijkstra in PHP

This is a clean, didactic version. It uses a simple linear scan for the minimum to keep it readable. For production, use a binary heap or Fibonacci heap.

 1// Dijkstra's Algorithm — simple, highly commented version
 2$graph = [
 3    'A' => ['B' => 2, 'C' => 2],
 4    'B' => ['C' => 1, 'D' => 3],
 5    'C' => ['D' => 2, 'E' => 3],
 6    'D' => ['E' => 1],
 7    'E' => []
 8];
 9 
10$source = 'A';
11$INF = 1_000_000;
12 
13// 1) Distance init
14$dist = [];
15foreach ($graph as $node => $_) $dist[$node] = $INF;
16$dist[$source] = 0;
17 
18// 2) Finalized set
19$visited = [];
20 
21// 3) Repeatedly extract the closest unvisited node and relax edges
22while (count($visited) < count($graph)) {
23    // extract-min
24    $minNode = null; $minDist = $INF;
25    foreach ($dist as $node => $d) {
26        if (!isset($visited[$node]) && $d < $minDist) {
27            $minDist = $d; $minNode = $node;
28        }
29    }
30    if ($minNode === null) break; // no more reachable nodes
31 
32    // finalize
33    $visited[$minNode] = true;
34 
35    // relax neighbors
36    foreach ($graph[$minNode] as $nbr => $w) {
37        if (!isset($visited[$nbr])) {
38            $alt = $dist[$minNode] + $w;
39            if ($alt < $dist[$nbr]) $dist[$nbr] = $alt;
40        }
41    }
42}
43 
44// 4) Print
45echo "Dijkstra distances from $source:\n";
46foreach ($dist as $node => $d) echo "$node: $d\n";

What to notice

• The algorithm always orders the frontier to pick the absolute smallest.
• That global ordering is where the log n overhead comes from.

Practical example 2: Batch-style deterministic idea in PHP

The full research algorithm is involved. The script below captures the core flavor: limited-depth relaxations from a frontier as a batch, then move the frontier forward. It avoids a global extract-min. It is meant for learning and mirrors the high-level design decisions.

 1// Batch-style multi-source relaxations — educational approximation
 2$graph = [
 3    'A' => ['B' => 2, 'C' => 2],
 4    'B' => ['C' => 1, 'D' => 3],
 5    'C' => ['D' => 2, 'E' => 3],
 6    'D' => ['E' => 1],
 7    'E' => []
 8];
 9 
10$source = 'A';
11$INF = 1_000_000;
12 
13// Distances
14$dist = [];
15foreach ($graph as $node => $_) $dist[$node] = $INF;
16$dist[$source] = 0;
17 
18// Limit per batch — think "k-hop" relaxations
19$k = 2;
20 
21// Start from the source frontier
22$frontier = [$source];
23$changed = true;
24 
25while ($changed) {
26    $changed = false;
27 
28    // If frontier is empty but improvements may exist, restart from all nodes
29    if (empty($frontier)) $frontier = array_keys($graph);
30 
31    // Track nodes improved in this batch
32    $touched = [];
33 
34    // Up to k layers from current frontier
35    $wave = $frontier;
36    for ($round = 1; $round <= $k; $round++) {
37        $nextWave = [];
38        foreach ($wave as $u) {
39            foreach ($graph[$u] as $v => $w) {
40                $alt = $dist[$u] + $w;
41                if ($alt < $dist[$v]) {
42                    $dist[$v] = $alt;
43                    $touched[$v] = true;
44                    $nextWave[] = $v;
45                    $changed = true;
46                }
47            }
48        }
49        if (empty($nextWave)) break;
50        $wave = $nextWave;
51    }
52 
53    // New frontier only where progress happened
54    $nextFrontier = [];
55    foreach (array_keys($touched) as $u) {
56        foreach ($graph[$u] as $v => $_w) $nextFrontier[$v] = true;
57    }
58    $frontier = array_keys($nextFrontier);
59}
60 
61// Print
62echo "Batch-style distances from $source:\n";
63foreach ($dist as $node => $d) echo "$node: $d\n";

What to notice

• No priority queue.
• Work happens in short waves from a reduced frontier, not by a global strict order every step.
• This matches the paper’s idea of bounded multi-source relaxations and frontier reduction, without reproducing all the complexity.

Why the new idea helps

Dijkstra pays a log n cost in selecting the next node because it maintains a full order of candidate distances. The new method avoids that by:

1. grouping work into batches,
2. limiting relaxations to k layers per batch,
3. shrinking the frontier to a small set of pivots before recursing. This breaks the sorting bottleneck and yields O(m · (log n)^(2/3)) time.

Complexity comparison

Rule of thumb

• Small or medium graphs: Dijkstra is simple and often fast enough.
• Very large directed graphs: the new approach has better asymptotic behavior and is theoretically faster.

FAQ

Does the new algorithm replace Dijkstra in everyday code Not necessarily. Dijkstra is simple and widely available. The new method is deeper and shines at scale.

Can the new idea handle negative weights The result is for non-negative weights in the comparison-addition model. Negative weights are a different problem with separate lines of research.

Final takeaways

• Dijkstra remains a practical baseline for non-negative weights.
• The 2025 deterministic SSSP shows we can beat the classic n log n barrier by changing how we explore the frontier. It uses bounded multi-source relaxations, a shrunken frontier, and recursion to avoid repeated global sorting.

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