Intermediate · Module 16 · Lesson 4
Jump & reachability
Track the farthest achievable frontier.
15–25 min readLesson 92 of 123
Lesson 92 of 123
1d streakTheory lesson · Step-by-step
Understand Jump & reachability
Track the farthest achievable frontier.
Worked approach
Trace the reasoning step by step
- 1State the candidate ordering and local choice made by Jump & reachability.
- 2Identify the feasibility condition maintained by the partial solution.
- 3Use an exchange or stays-ahead argument to justify the choice.
- 4Commit the choice and remove candidates it makes impossible.
- 5Search for a counterexample before trusting the greedy rule.
Concrete example
Can [2,3,1,1,4] reach the last index?
- 1From index 0, farthest is 2.
- 2Index 1 extends farthest to 4.
The last index is reachable.
What must remain true (Loop Invariant)
After every choice in Jump & reachability, the partial solution can still be extended to an optimum under the proved exchange or stays-ahead argument.
Cost Model
Complexity & Trade-offs
OperationTimeSpaceWhy
Farthest-frontier scan
O(n)
O(1)
The bound follows from the number of states visited and the work performed per transition.When to use it
- Scheduling, reachability, coding, matching resources, and minimum-cost growth.
- Problems with an exchange proof or matroid-like safe choices.
- Online decisions that maintain a best frontier.
Common mistakes
- Choosing a plausible local optimum without a proof.
- Using the wrong sorting key.
- Applying greedy reasoning when choices interact through future state.
Practice prompts
- Reproduce the worked Jump & reachability example without looking at the trace.
- Change one input condition and explain whether the Jump & reachability invariant still holds.
- Implement the approach, derive its cost, and compare it with a straightforward baseline.
Mastery check
Can you derive it instead of memorizing it?
State the invariant before looking at code.
Explain why every transition is safe.
Construct a worst-case input.
Name one alternative and its trade-off.