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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2017/functional-programming-design-and-analysis-of-algorithms-resit/5/a/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2017 / Functional programming, design and analysis of algorithms (resit) / 5 / a / Solution by

Codex  0 2026-10-31

Initialize

d [s] = 0

and every other tentative distance to infinity. The Bellman-Ford algorithm repeatedly relaxes every directed edge:

for v in V: d[v] = infinity; parent[v] = none
d[s] = 0
repeat |V|-1 times:
    changed = false
    for each edge (u,v) of weight w:
        if d[u] != infinity and d[u] + w < d[v]:
            d[v] = d[u] + w
            parent[v] = u
            changed = true
    if not changed: break
for each edge (u,v) of weight w:
    if d[u] != infinity and d[u] + w < d[v]:
        report a reachable negative-weight cycle

An unreachable vertex retains infinity. A reachable negative-weight cycle prevents finite shortest-walk distances for vertices reachable from that cycle, so the detection case must not be reported as an ordinary finite answer. With edge-list scans the worst-case work is $O (∣ V ∣∣ E ∣ + ∣ V ∣)$ , conventionally stated as

O (∣ V ∣∣ E ∣)

, and the auxiliary storage is

O (∣ V ∣)

beyond the graph.

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