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Early and late event times, critical path

The forward and backward passes find how long a project takes and which activities cannot be delayed.

Forward pass

Set the source's early event time to 0. For each event in order, look at every arc into it: early time of its start event + its duration. The early event time is the largest of these, because every activity in must be finished. The early time of the sink is the minimum project duration.

Backward pass

Set the sink's late event time equal to its early time. Working backwards, for each event look at every arc leaving it: late time of its end event - its duration. The late event time is the smallest of these.

Critical activities

An activity from event i to event j is critical if early(i) = late(i), early(j) = late(j) and late(j) - early(i) = its duration. A critical path is a path of critical activities from source to sink; there may be more than one. An activity joining two critical events is not automatically critical: check the duration.

Presentation

Write each event's early and late times in a box at the node, left box early and right box late (as the question shows). Include dummies with duration 0.

Worked example

Find the early and late event times for project J and the critical path.

  1. Early: event 0 0, event 1 4, event 2 7, event 3 11, event 4 14, event 5 13, event 6 19
  2. Late: event 0 0, event 1 4, event 2 7, event 3 14, event 4 14, event 5 16, event 6 19
  3. Critical activities: A, C, F, H

Answer: Project time 19 days; critical path A C F H

Key idea: Forward pass takes the largest, backward pass the smallest. Critical: both ends critical and the duration fills the gap exactly.

Check yourself

  1. Find the minimum completion time of project J.
    Show answer

    19 days. Early time of the sink (event 6) = 19.

  2. Find the early event time of event 3 in project J (the end of D and E).
    Show answer

    11. Into event 3: D gives 4 + 5 = 9; E gives 7 + 4 = 11. Largest: 11.

  3. Find the late event time of event 3 in project J (the start of G).
    Show answer

    14. Out of event 3 only G: late(5) - 2 = 16 - 2 = 14.

  4. Find the late event time of event 2 in project J (the start of E and F).
    Show answer

    7. E: 14 - 4 = 10; F: 14 - 7 = 7. Smallest: 7.

  5. Which activities of project J are critical?
    • A, C, F, H
    • B, F, H
    • A, D, G, I
    • A, C, E, G, I
    Show answer

    A, C, F, H. Activities with zero float: A, C, F, H.

  6. In the forward pass, why is the largest value taken at each event?
    • To make the project as long as possible
    • Because Dijkstra takes the largest value
    • Every activity into the event must finish before activities leaving it can start
    • To find the float
    Show answer

    Every activity into the event must finish before activities leaving it can start. The event cannot happen until the last of its incoming activities finishes.

  7. Two events i and j are both critical. Is the activity from i to j always critical?
    • No: it is only critical if its duration equals late(j) - early(i)
    • Yes, always
    • Only if it is the longest activity
    • Only if it ends at the sink
    Show answer

    No: it is only critical if its duration equals late(j) - early(i). An activity between critical events can be shorter than the gap, so it has float.

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Last updated 7 November 2026. Spotted a mistake? Tell us.