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A greedy algorithm builds a feasible solution by repeatedly committing to a locally preferred next choice. Each commitment is final: the algorithm does not revisit it when later choices become available. In mathematical optimization, the important question is whether those commitments can still attain the best objective function value.

For example, a shared telescope can accept several observation requests with fixed time slots. In interval scheduling, every completed request counts equally, and overlapping requests cannot both be accepted. Choosing the request that finishes earliest leaves the resource available sooner. The example below turns that intuition into a precise rule, then shows why choosing the earliest start or shortest duration can lose jobs.

Greedy algorithms are attractive because a single pass can replace a search through many candidate subsets. However, "locally best" is only a design proposal. A problem-specific greedy-choice property, often established by an exchange argument, must connect the local decision to the actual objective function. The largest-individual-value rule in the 0-1 knapsack problem fails; Dijkstra's algorithm needs its nonnegative-edge assumptions. A rule's success in one optimization task does not justify it in another.

A greedy-choice property says that some optimal feasible solution contains the locally selected choice. Together with optimal substructure in the remaining instance, it allows repeated commitments. An exchange argument often establishes this property.^[ref]

For interval scheduling, a safe first choice need not belong to every optimal feasible solution. When several optima exist, a commitment may exclude one while retaining another. The useful loop invariant is that the accepted prefix can still be extended to an optimum. Merely keeping the accepted intervals compatible proves feasibility, which is a weaker claim.

A divide-and-conquer algorithm splits an instance into smaller instances of the same kind, solves them recursively, and combines their answers. The smaller instances should be sufficiently smaller to ensure algorithm termination; the combination cost determines whether splitting saves work. The two halves of merge sort illustrate this separation.

Dynamic programming identifies a collection of subproblems and dependencies among them, solves each distinct subproblem once, and stores its answer. A table in dependency order or memoization prevents repeated evaluation of overlapping subproblems. For optimization, an optimal substructure argument justifies constructing an optimal answer from optimal smaller answers.

A greedy algorithm repeatedly makes a locally best feasible choice and commits to it. It avoids exploring alternative continuations. Algorithm correctness therefore needs a greedy-choice property, often proved by replacing the first choice of an optimal solution with the greedy choice without worsening it.

Divide-and-conquer algorithms suit problems with largely independent, balanced subproblems and an inexpensive combination step. In merge sort, the two halves are independently sorted and merged in linear time. Its recurrence relation is $T(n)=T(\lfloor n/2\rfloor )+T(\lceil n/2\rceil )+\Theta (n)$, giving $\Theta (n\log n)$ time complexity.

Dynamic programming suits overlapping subproblems with a compact state description. For the coin change problem, let $C(v)$ be the minimum number of coins needed to make amount $v$. With positive denominations $D$ containing $1$,Every optimal nonempty answer has a last coin $d$ and an optimal answer for $v-d$; otherwise replacing the remainder improves it. Filling $C(0),\dots ,C(V)$ takes $O(V|D|)$ time complexity and $O(V)$ space complexity. Repeated amounts are computed just once. This bound is in the numeric amount $V$, not its binary input length.

A greedy algorithm suits problems where one can prove a greedy-choice property. For unweighted interval scheduling, choose an interval with earliest finishing time, discard overlapping intervals, and repeat. Among optimal schedules, replace the first interval by the earliest-finishing one. The replacement finishes no later, so every later interval remains feasible. Applying the same argument to the remaining intervals proves optimality. After sorting by finish time, a single scan suffices.

These examples separate the relevant structural reasons: independent halves, repeated amount states, and an exchange argument certifying each irreversible choice.

In the coin change problem, take denominations $\{1,3,4\}$ and target amount $6$. The natural greedy algorithm repeatedly chooses the largest denomination not exceeding the remaining amount. It chooses $4$, then $1$, then $1$, for three coins. Yet $3+3=6$ uses only two.

Two is optimal: one coin cannot make $6$ because denomination $6$ is absent, and the displayed pair attains the lower bound. ThereforeThe locally largest first coin loses the possibility of using the better pair of threes. This supplies a concrete failure of the greedy-choice property; optimal substructure alone does not justify the greedy choice. The dynamic programming recurrence from part (b) would compare all feasible first or last denominations and find the pair.

In this separate three-job example, labels $A,B,C$ denote new jobs:Their durations are five, five and two. Consider the rule "accept a shortest remaining job, discard all jobs conflicting with it, and repeat". It selects $C$ first. But $C$ overlaps $A$ on $[4,5)$ and $B$ on $[5,6)$, so both are discarded. The rule returns one job, while the compatible subset $\{A,B\}$ contains two. The common endpoint five is allowed, and no feasible solution contains all three jobs, so two is the maximum.

Duration alone ignores where a job sits. Here the shortest job straddles the useful handover time. Earliest finish instead chooses $A$, then $B$, and attains two. A counterexample can refute a proposed greedy-choice property even when the rule succeeds on many other inputs.