Early path finalization repair: keep the negative edge

A plausible intermediate score can become wrong when a procedure declares it permanent too soon. Early path finalization repair starts by recording the exact procedure, the unchanged input and a complete route that contradicts the result. In this original paper exercise, a four-vertex directed graph makes the mistake visible. The failed procedure returns four, although an allowed route scores three. Processing vertices in a dependency order repairs this example without deleting its negative edge or changing what the scores mean.

Write the whole graph before tracing the procedure

Use four vertices named S, A, B and T. S is the start and T is the destination. There are exactly five directed edges: S to A has score two; S to B has score five; B to A has score negative four; A to T has score two; and B to T has score one hundred. There are no other edges. A route follows arrow directions, and its score is the sum of its edge scores. The task is to find the smallest score among complete routes from S to T.

These are stipulated integers in a mathematical model. A score is neither geometric distance nor travel time, money, water flow or a physical product measurement. In particular, negative four is an allowed contribution to the sum; it does not describe water moving backward or a negative length. Keeping that meaning explicit prevents a familiar diagram from importing assumptions the exercise never supplied.

The Sunnydaze DW-34112 fountain account supplies separate catalog context. Its twelve originals show a five-tier rock-style fountain, including whole views, a back opening and marketing panels. One customer photograph adds fairies, flowers and a sign; those are not adopted as supplied contents. Dimensions, quietness, lighting and relaxation claims remain unverified listing statements. The pictured cascades provide none of the graph's edges or scores.

Define the failing procedure exactly

Give S a tentative score of zero. Leave the other three tentative scores unset initially. Repeatedly select the unsettled vertex with the smallest currently assigned tentative score, mark that selected vertex permanently settled, and inspect its outgoing edges. For an unsettled neighbor, form a candidate by adding the edge score to the selected vertex's score. Assign the candidate if the neighbor was unset, or replace its tentative score if the candidate is smaller. Never change a settled vertex.

Stop when T is selected and return its score. This stopping rule and the refusal to change settled vertices belong to the procedure being tested. They cannot be supplied afterward to explain away a disappointing result. There are no tied selections in the trace below, so a tie-breaking rule cannot account for the failure. Write a separate settled list beside the tentative scores to distinguish what the procedure knows from what it has prohibited itself from changing.

First select S with score zero. Its outgoing edge to A assigns A the tentative score two. Its outgoing edge to B assigns B the tentative score five. T remains unset. S is now settled. Among the remaining assigned vertices, A has the smaller score, so A is selected next and permanently settled at two.

From A, the edge to T adds two and assigns T the tentative score four. The remaining assigned unsettled scores are T at four and B at five. The procedure therefore selects T next, settles it at four, stops and returns four. Its selected sequence is S, A, T. The returned route S to A to T is allowed, and its sum two plus two really is four. The error concerns minimality, rather than whether that route exists.

Show the missed route before proposing a fix

The route S to B to A to T is also allowed. Its score is five plus negative four plus two, which equals three. Three is smaller than the returned four. That single concrete witness is enough to disprove the procedure's answer for this input. No assertion about a large graph collection is necessary to establish the failure of this exact run.

The unprocessed vertex B has tentative score five, larger than A's two. But B's outgoing edge to A contributes negative four. Reaching A through B therefore gives five minus four, or one. A's previously settled value two was not its smallest attainable score. The procedure chose to make it permanent before examining an incoming contribution that could improve it.

Notice two related restrictions in the failed procedure. It stops at T before processing B at all, and it refuses to revise A once A is settled. Simply saying “continue longer” leaves the latter restriction unresolved. Even if B were processed after the reported stop, a procedure that still rejects updates to settled A would discard the improving score one. A useful repair must identify which processing dependency will change, rather than merely request more steps.

Keep the scores and change the processing order

For this graph, use the order S, B, A, T. Every arrow leads from an earlier vertex in that order to a later one: S precedes A and B; B precedes A and T; and A precedes T. Thus each vertex is processed after all its possible incoming predecessors have already been processed. The order follows the graph's dependencies, rather than selecting the smallest currently visible score.

Begin again from the preserved input, with S zero and the other scores unset. Processing S produces A at two and B at five, just as before. The next vertex is B because of the declared dependency order, even though its current score is larger than A's. Its edge to A produces candidate one, improving A from two to one. Its edge to T produces candidate one hundred five, assigning that value to T.

Now process A at one. Its edge to T produces candidate three. That is smaller than T's current one hundred five, so replace T's score with three. Finally process T at three. It has no outgoing edge. The resulting route can be reconstructed from the improving predecessors: T came through A, A came through B, and B came from S. Read forward, that is S, B, A, T.

No edge score changed during this repair. B to A still contributes negative four, and the direct B to T edge still contributes one hundred. The repaired record answers the same minimum-score question as the original input. Keep the original failed trace beside it so another reader can see precisely why choosing the next smallest tentative score and choosing the next dependency-ready vertex are different decisions here.

Certify the answer with every complete route

There are exactly three complete routes from S to T. If the first edge goes to A, A has only the outgoing edge to T, producing S, A, T with score four. If the first edge goes to B, B can go directly to T, producing S, B, T with score one hundred five. Alternatively, B can go to A and then T, producing S, B, A, T with score three. Those choices exhaust the outgoing edges available at every stage.

The minimum of four, one hundred five and three is three. This complete list independently certifies the repaired result; it does not merely repeat the new procedure's final label. The graph has no directed cycle, so there is no way to repeat a loop to create another route. A diagnosis involving a negative cycle would describe an object absent from the input. The negative edge alone explains the failed permanent settlement.

The exact-total minimum-step exercise compares repeated positive moves using the cost of every eligible last move. Its states are accumulated totals and its output counts moves. This graph instead has signed edge scores and a premature-settlement error. Sharing the word minimum does not make the two state spaces, failure mechanisms or certificates interchangeable.

Reject a change that quietly answers another question

Replacing negative four with zero may look like a convenient adjustment. It changes the three route scores to four, one hundred five and seven. The new minimum is four. A procedure returning four on that altered graph has not repaired its answer for the original graph, whose minimum remains three. Save the modified edge explicitly if you want to study that second exercise; do not label it an unchanged-input recovery.

The same distinction applies to deleting the B to A edge. That removes the winning route rather than accounting for it. Neither alteration is justified by the task's score definition. The relevant test is whether the repair preserves the original allowed routes and their sums while making the answer correct. A convenient result after changing the task cannot supply that test.

The complete-equation swap repair preserves two equations while changing their order, then checks the answer in the originals. Here the preserved object is a directed graph and the final certificate is a complete route list. An equation's solution-set argument does not itself prove a graph-processing rule, but keeping an untouched original is useful in both separate exercises.

Use controlled variations without overstating the result

A small exact family checks that the failure is not tied to the particular labels two and five. Set the S to A score to a, set S to B to a plus g, and set A to T to d. Let B to A contribute negative g minus one, while B to T remains one hundred. Choose a from one through four, g from two through five, and d from one through g minus one.

The failed procedure selects A at a, then T at a plus d because d is smaller than g. But the route through B reaches A at a minus one, and reaches T at a minus one plus d. It improves the reported result by exactly one. The direct route through B scores a plus g plus one hundred and cannot win in these stated ranges. There are forty combinations, and exact integer controls reproduce that strict gap in all forty.

These controls help catch a missed minus sign or an incorrectly implemented stopping rule. The explicit three-route enumeration supplies the decisive certificate for the original example. Neither forty passing controls nor this four-vertex repair establishes a solver for every directed graph. A different input can require a different argument about dependencies, reachability or cycles. This article also supplies no measured runtime, memory result or claim about numerical rounding.

The Alpine WIN1334 rainforest fountain account remains catalog context. All eleven originals show the same three-tier complete fountain through whole views, closeups and staged marketing panels. The native account gives differing horizontal dimensions; neither is adopted as a verified measurement. Materials, sound, electrical suitability, warranty and maker identity are not authenticated. Its depicted streams do not define the paper model's processing order.

A reader considering a real item order on republeq.com should examine the particular offer and checkout independently of this directed-graph exercise. The assigned values are invented integers, not prices, payment amounts or evidence about a merchant. If USDT is being considered, first verify whether the current checkout actually offers that cryptocurrency method for the intended purchase. An article mention provides no availability evidence. Before authorizing any transfer, clarify the requested asset, network, amount, receiving destination and order reference with the responsible service. Review the actual item identity, quantity, delivery details, total charge and applicable terms for that order. Any fee, conversion term or exchange rate requires current evidence of its own; none is supplied by the edges or the route comparison. Determine the applicable cancellation, return and refund process before relying on one, including how a refund would be handled for the selected payment method. Keep the relevant instructions and transaction records, including dates and confirmed references. Ask the responsible service to confirm any essential changed instructions. Retain any acknowledgement with the exact order details. If a payment or order status remains uncertain, reconcile that status with the responsible service before considering another transfer. A selected route does not show that an order was accepted, and a correct value calculation cannot verify a receiving destination. This exercise supplies no wallet address, transfer guarantee, recovery procedure or assessment of a counterparty. Completing the mathematical worksheet requires no purchase and no payment. Decide whether to proceed only after the relevant conditions for the particular transaction are clear, and keep its records separate from the invented edge scores and processing record.

Retain the failure and the original-input certificate

Keep the five directed edges, the failed S, A, T trace and the returned four together. Attach the improving S, B, A, T witness with score three. Then retain the dependency order S, B, A, T and the complete list of three route scores. This record identifies both the rejected inference and the verified repair: a larger unfinished prefix can still improve an earlier vertex through a negative contribution, so this example must not finalize that vertex before its incoming dependencies have been processed.