Weighted route cost mismatch: check the full sums

Weighted route cost can disagree with a count of edges. In the fictional directed graph here, the direct path from A to D uses one edge and costs eight units. Going through B uses two edges and costs four. A worksheet that calls the direct path cheapest because it has fewer steps has compared the wrong quantity.

All vertices, edges and costs below are invented mathematical inputs. The graph is not a travel map, delivery network, shopping comparison or payment system. Its positive weights are abstract cost units, without a currency or measured distance. The task is to repair a comparison by retaining the stated objective: minimize the complete path's sum of edge weights.

Write every directed edge

Use four vertices named A, B, C and D. The complete directed edge list is A to D with weight 8, A to B with weight 2, A to C with weight 1, B to D with weight 2, and C to D with weight 10. There are no other edges in this exercise.

A directed edge permits movement only in its printed direction. Listing A to B does not also create B to A. Likewise, placing B and C beside one another on a sketch does not create a connection between them. Keep the five-edge list beside the drawing so that layout does not silently add another permitted step.

Every candidate must start at A and finish at D. D has no outgoing edge. B and C each have exactly one outgoing edge, leading to D. These small structural facts make a complete enumeration possible. They also prevent a partial path ending at B or C from being accepted as a completed answer.

List the three complete paths

From A, the first edge must lead to D, B or C, because these are its only three outgoing choices. Choosing A to D reaches the destination immediately. This gives the first complete path, written A-D. It uses the single edge whose stipulated weight is eight.

Choosing A to B leaves the path at B. Its only next edge is B to D, so this branch gives A-B-D. There is no choice to move from B to C, remain at B for another step, or return to A. Those actions are absent from the declared edge list.

The remaining first choice leads to C. Its only next edge is C to D, giving A-C-D. We have now covered every first edge from A and every continuation each branch permits. Thus A-D, A-B-D and A-C-D are the complete set of allowed A-to-D paths in this graph.

This completeness argument matters when claiming a minimum. Merely comparing three convenient paths would leave open whether a fourth permitted path had a smaller cost. Here the edge list and the forced continuations establish why there is no fourth path to check. The drawing's apparent simplicity is not being used as that proof.

Keep edge count and cost in separate columns

For A-D, record edge count one and total cost eight. The cost is the weight of its sole edge. The two vertex names are not two edges; one permitted transition joins them. Keeping the starting vertex in the written path helps identify the route without increasing its edge count.

For A-B-D, record edge count two. Add the weight of A to B, which is two, to the weight of B to D, also two. The complete sum is 2 + 2 = 4. Both transitions belong in the cost record, including the final step into D.

For A-C-D, the edge count is also two. Its weights are one and ten, producing 1 + 10 = 11. The two two-edge paths therefore have different costs. A column containing only their common edge count could not distinguish the complete totals four and eleven.

Read each cost back against its own path. A row labelled A-B-D but showing 1 + 10 would contain C's edge weights under B's vertex sequence. Repair that copying error before ranking the rows. A correct sum detached from the edges that supplied it would describe another path.

Identify the minimum using complete totals

The three complete costs are eight, four and eleven. Four is smaller than both eight and eleven, so A-B-D has the minimum cost under the stated weights. It is the unique minimum among this complete three-path set because neither other path has the same total.

The proof has two parts: the path list is complete, and its recorded sums have been compared. Finding A-B-D with cost four alone would demonstrate a path costing four. Without the completeness check, that discovery would not establish that no cheaper permitted path remained unexamined.

A-C by itself has cost one, but it stops at C. It is a prefix of a complete path, not an eligible A-to-D answer. Its remaining required edge costs ten. Comparing that unfinished cost of one with the complete cost of four would omit both the destination requirement and an applicable edge weight.

Diagnose the fewest-edge proposal

The direct A-D path genuinely uses the fewest edges: one rather than two. That statement is correct for this graph. The error occurs when the worksheet transfers the result into a claim about the smallest cost. Edge count and total weight are distinct columns, and the supplied weights do not make them interchangeable.

Do not change the direct path's edge count to make the cost ranking look consistent. It still has one edge even though its cost exceeds A-B-D's. A truthful repair retains both observations: A-D minimizes the number of edges, while A-B-D minimizes the sum of the declared weights.

Similarly, a drawn segment's physical length supplies no additional cost here. The graph has no coordinate scale or rule turning centimetres on the page into weight. Making the direct edge shorter in the sketch would change its appearance, while its written weight would remain eight.

Test the cheapest-first-edge proposal

Consider another proposed rule: at each vertex, take its outgoing edge with the smallest displayed weight. At A, this rule chooses A to C, since one is smaller than two and eight. There is no tie at that first choice, so the proposal does not need a tie-breaking convention in this example.

At C, the only outgoing edge leads to D and costs ten. The resulting complete path is A-C-D with cost eleven. It is more expensive than both the direct path's eight and A-B-D's four. Choosing the cheapest first edge has therefore failed to minimize the complete sum in this particular graph.

The missing information was already present in the edge list. The first weight of one did not include the later ten. This diagnosis does not claim that every local-choice procedure fails in every graph, and it supplies no implementation or performance result for a named search algorithm. It checks the precise proposed rule against these five edges.

Use a separate equal-weight control

On a separately labelled copy, replace every edge weight with one while retaining the same directed connections. The three path costs become one, two and two. In this changed model, each edge contributes exactly one unit, so a path's cost equals its edge count.

The direct path is then the minimum-cost choice. This control explains the condition that makes edge counting appropriate here. It does not validate using edge count with the original weights of eight, two, one, two and ten. Keep the equal-weight copy distinct rather than moving its conclusion back onto the earlier record.

Change one weight without changing the graph

Make another separate version in which only the direct A-to-D weight changes from eight to three. Retain the other four original weights. The path list and edge counts remain unchanged, while the complete costs become three for A-D, four for A-B-D and eleven for A-C-D.

The direct path is now cheapest because three is the smallest complete total. The change illustrates why the actual weights belong beside a minimum-cost conclusion. The same connection pattern can support a different winner after an explicitly changed input; it does not make either correctly labelled calculation inconsistent.

Leave a checkable repaired record

Keep the original five directed edges, the A-to-D endpoint requirement and the sum-of-weights objective at the top. Under them, retain all three paths with separate edge-count and cost columns. Mark the original cost-four A-B-D result beside the comparisons that establish it.

Preserve the rejected proposals with their exact failures. The fewest-edge proposal correctly finds one edge but answers another objective. The cheapest-first-edge proposal follows A-C-D and totals eleven. The incomplete A-C proposal fails to reach D. These are different errors, so each needs its own reason rather than one unexplained cross beside the worksheet.

Read optional workspace accounts separately

The native MLINS organizer account has one image showing a black mesh assembly with stacked trays and an integrated lower drawer. Books, a vase, foliage and furniture are staged scene objects, not verified supplied stock. The catalog label does not authenticate maker, material, dimensions, capacity, stability or organizing performance.

The native Prosumer's Choice organizer account has one image showing a brown wood-grain assembly with one wide upper drawer, two lower drawers and a shallow top shelf. Bamboo is retained as catalog wording without independent material verification. No maker, dimension, document-fit, capacity, durability or performance claim is authenticated here.

Neither account defines the fictional graph or supplies its weights. The exercise requires no purchase and no operation of either organizer. Keep actual contents and suitability questions attached to the product you are considering, separately from the route calculation.

Distinguish nearby graph questions

The one-position change puzzle counts unit replacements in fixed letter positions. Its lower bound and matching construction concern that movement rule. Here one graph edge can carry a different weight from another, so minimum edge count does not automatically give minimum complete cost.

The graph coloring on paper guide checks whether connected vertices receive different labels. This worksheet instead sums weights along directed A-to-D paths. Vertex colors, the number of available labels and nearby positions on a sketch do not supply the route-cost objective.

Separate any actual checkout enquiry

If you independently consider a desktop-organizer order on republeq.com and want to ask about USDT, begin with the exact offer and intended quantity. The vertices and weights in this worksheet are invented mathematical inputs, not product dimensions, prices, cryptocurrency amounts or payment instructions. Correct arithmetic establishes neither a currently offered payment method nor the status of an actual order. Read the choices and conditions presented for your intended transaction, and ask the responsible support contact to clarify essential missing information before deciding whether to proceed. If a cryptocurrency option is offered, establish the specified asset and network, requested amount, any stated validity period and the reference associating a payment with the order. Do not infer a transfer destination from a product photograph, a drawing or an earlier transaction. Resolve product and supplied-content questions separately from payment instructions. Ask about applicable delivery, cancellation, return and refund conditions, including the denomination of any approved refund and relevant deductions. Keep each answer attached to the offer it concerns. Information about one organizer account does not automatically describe the other, and a changed quantity may require reviewing the resulting order information again. This article promises no present USDT availability, fee level, exchange result, confirmation time, refund amount or delivery outcome. You can inspect every fictional path without purchasing anything. Preserve any eventual order acknowledgement and transaction record separately from this worksheet. Those records concern an actual transaction under its documented conditions; the path comparison follows expressly declared edges. A correct minimum-cost result cannot resolve an unanswered checkout question.