📚 Edexcel Further Maths D1 Decision Mathematics 1 Mark Scheme Essentials | 爱德思进阶数学决策数学 1 评分要点精讲
This revision guide covers the highest-impact content for Edexcel Further Maths Decision Mathematics 1. It is written as a mark scheme companion: every section highlights the method marks, working steps and common errors that examiners look for.
本复习指南涵盖爱德思进阶数学决策数学 1 中最核心的考点。它是一份评分要点伴读指南：每一节都突出考官关注的方法分、书写步骤和常见错误。
1. Algorithms and Complexity | 算法与复杂度
In D1 exam questions, an algorithm is a finite sequence of precise instructions. You must be able to trace an algorithm given in a flow chart or pseudo-code and state the output at each stage.
在 D1 考试题中，算法是一组有限的精确指令。你必须能够对流程图或伪代码给出的算法进行逐步跟踪，并在每个阶段写出输出结果。
Mark schemes award method marks for showing each pass, iteration or comparison. Never write only the final answer; record the working line by line, because even a small arithmetic error can still earn most of the marks if the process is visible.
评分方案对展示每一轮、每一次迭代或每一次比较给方法分。不要只写最终答案；要逐行记录过程，因为即使出现小的算术错误，只要过程清晰可见，仍能获得大部分分数。
Common command words include ‘apply’, ‘state the order’, and ‘explain why the algorithm is finite’. An algorithm is finite if it has a stopping condition and does not loop forever.
常见指令词包括“应用”“写出次序”以及“解释为什么该算法是有限的”。如果算法有终止条件且不会无限循环，那么它就是有限的。
When comparing algorithms, examiners expect you to refer to the number of steps, comparisons or swaps, and not only to say that one is ‘faster’.
在比较算法时，考官希望你提到步数、比较次数或交换次数，而不仅仅是说某一个“更快”。
2. Sorting and Searching | 排序与搜索
Bubble sort works by repeated comparisons of adjacent items, swapping them if they are in the wrong order. After the first pass the largest value is in its final position, so later passes can ignore it.
冒泡排序通过反复比较相邻元素并交换错误顺序进行。第一轮之后最大值到达最终位置，因此后续轮次可以忽略它。
In quick sort, choose a pivot, often the middle value, then place values less than the pivot to the left and values greater than the pivot to the right. Repeat on each sublist until every item has been used as a pivot or the sublist has length one.
快速排序选择一个枢轴，通常是中间值，然后将小于枢轴的值放到左边，大于枢轴的值放到右边。对每个子列表重复，直到每个元素都作过枢轴或子列表长度为 1。
A mark scheme question often asks for the number of comparisons or swaps. Set out passes clearly, underline swapped items, and state when the list is sorted or when no further swaps are needed.
评分题经常问比较次数或交换次数。清晰列出每一轮，标出被交换的元素，并说明列表何时已经排好或不再需要交换。
Binary search requires an ordered list. Compare the target with the middle item; if it is smaller, discard the upper half, and if it is larger, discard the lower half. Repeat until the item is found or the search interval is empty.
二分搜索要求列表有序。将目标与中间项比较；如果目标更小，舍弃上半部分；如果目标更大，舍弃下半部分。重复直到找到目标或搜索区间为空。
3. Bin Packing | 装箱问题
First-fit places each item into the first bin that has enough remaining space. You must process items in the given order and record the contents of every bin after each decision.
首次适应法将每个物品放入第一个有足够剩余空间的箱子。你必须按给定顺序处理物品，并在每次决策后记录每个箱子的内容。
First-fit decreasing sorts the items into non-increasing order before applying first-fit. Mark schemes often give credit for a correct sorted list even if the final packing is slightly wrong.
降序首次适应法先将物品按非递增顺序排列，再使用首次适应法。即使最终装箱略有错误，评分标准通常也会对正确的排序列表给分。
Full-bin packing looks for combinations that exactly fill a bin. It is a heuristic and does not always give the optimal number of bins, but it can produce useful upper bounds.
满箱装箱寻找恰好填满箱子的组合。它是一种启发式算法，不一定给出最优箱数，但可以产生有用的上界。
When a question asks for a lower bound, use the sum of item sizes divided by bin capacity, rounding up to the next integer. State this calculation explicitly.
当题目要求下界时，用物品总大小除以箱子容量并向上取整。要明确写出这一计算过程。
4. Graphs and Networks | 图与网络
A graph is a set of vertices connected by edges; a network is a weighted graph. You must be confident with terms such as degree, path, cycle, tree, connected and subgraph, because paper questions often test definitions directly.
图是由边连接的一组顶点；网络是带权重的图。你必须熟悉度、路径、圈、树、连通图和子图等术语，因为试卷经常直接考查定义。
A tree is a connected graph with no cycles. A spanning tree of a graph connects all vertices without creating a cycle, so it has exactly n − 1 edges for n vertices.
树是连通且无圈的图。生成树连接所有顶点并且不形成圈，因此对于 n 个顶点，它恰好有 n − 1 条边。
In mark schemes, definitions must be precise. Do not say ‘a tree has no loops’ when you mean cycles, and do not confuse a path with a trail or an Eulerian graph with a Hamiltonian graph.
评分标准中定义必须准确。当你指“无圈”时不要说成“无环”，也不要混淆路径与迹、欧拉图与哈密顿图。
A connected graph has a path between every pair of vertices. A complete graph has every pair of vertices joined by exactly one edge.
连通图中任意两个顶点之间都存在路径。完全图中每一对顶点之间恰好由一条边相连。
5. Minimum Spanning Trees | 最小生成树
Kruskal’s algorithm: list all edges in ascending order of weight, add an edge if it does not form a cycle, and continue until all vertices are connected. Record rejected edges and the reason for rejection.
克鲁斯卡尔算法：将所有边按权重递增排列，如果加入一条边不会形成圈则添加，继续直到所有顶点连通。记录被拒绝的边以及拒绝原因。
Prim’s algorithm grows a tree from a chosen start vertex by repeatedly adding the cheapest edge that connects a new vertex to the current tree. You must state your starting vertex at the beginning.
普里姆算法从选定起点生长一棵树，反复添加连接新顶点到当前树的最便宜边。你必须在开始时说明起点。
Mark schemes may accept either method, but you must not mix them without explanation. Show the order of edge selection and give the total weight clearly.
评分标准可能接受任一方法，但不能未加说明地混用。展示选边顺序并清楚给出总权重。
Use a table or a running list with columns for edge, weight and action. If you use Prim, a small table of nearest distances from the growing tree can prevent missed edges.
使用表格或起列清单，包含边、权重和处理方式。如果使用普里姆算法，一个小表格记录当前树到各顶点的最近距离可以防止漏边。
6. Dijkstra’s Shortest Path | Dijkstra 最短路径
Dijkstra’s algorithm finds the shortest path from a start vertex to all other vertices by updating temporary labels. At each step, choose the unvisited vertex with the smallest working value, make it permanent, and update its neighbours.
Dijkstra 算法通过更新临时标号找出从起点到所有其他顶点的最短路径。每一步选择工作值最小的未访问顶点，将其设为永久标号，并更新其邻点。
Exam questions require working values at every vertex, the order of permanent labelling, and the final route traced backwards from the target vertex.
考题要求写出每个顶点的工作值、永久标号的顺序，以及从目标顶点反向追溯得到的最终路线。
A common mark scheme penalty is giving only the shortest distance without any working labels. The method mark is often for the sequence of permanent labels, not for the final number alone.
一个常见的评分扣分点是只给出最短距离而不展示任何工作标号。方法分通常给的是永久标号顺序，而不仅仅是最终数字。
When several vertices have the same working value, either may be chosen next, but you should state your choice and keep the remaining working values unchanged.
当多个顶点具有相同的工作值时，可以任选其一，但应说明你的选择，并保持其余工作值不变。
7. Route Inspection (Chinese Postman) | 路由检查（中国邮递员）
The route inspection problem asks for the shortest route that traverses every edge at least once, starting and ending at the same vertex. If the graph is Eulerian, meaning every vertex has even degree, the optimal route length is simply the sum of all edge weights.
路由检查问题要求找出至少经过每条边一次、起点与终点相同的最短路线。如果图是欧拉图，即每个顶点的度都是偶数，那么最优路线长度就是所有边权重之和。
If there are odd-degree vertices, they must be paired so that the total extra distance is minimised. Identify all odd vertices, find shortest paths between every pair, and choose the pairing with the smallest total length.
如果存在奇度顶点，则必须将它们配对，使额外总距离最小。找出所有奇度顶点，计算每对之间的最短路，并选择总长度最小的配对。
Add the repeated edges corresponding to the chosen pairing to the total weight. State clearly which edges are repeated and list the odd vertices paired.
将与所选配对对应的重复边加到总权重上。清楚说明哪些边被重复，并列出配对的奇度顶点。
Mark schemes reward a systematic table of pairings and totals. If you simply write an answer with no pairing evidence, you may lose method marks.
评分标准奖励系统的配对及总距离表格。如果你只写一个答案而没有配对依据，可能会失去方法分。
8. Travelling Salesman Problem | 旅行商问题
The travelling salesman problem, or TSP, seeks a Hamiltonian cycle of minimum total weight that visits every vertex exactly once and returns to the start. It is an NP-hard problem, so exam questions use heuristics and bounds rather than complete enumeration for large graphs.
旅行商问题，即 TSP，要求找到一个总权重最小的哈密顿圈，恰好访问每个顶点一次并回到起点。它是一个 NP 难问题，因此对于较大的图，考试题使用启发式算法和上下界而非完全枚举。
The nearest neighbour algorithm starts at a chosen vertex and repeatedly goes to the nearest unvisited vertex, finally returning to the start. It is fast but not guaranteed to be optimal.
最近邻算法从选定顶点出发，反复前往最近的未访问顶点，最后回到起点。它计算很快，但不能保证最优。
To find an upper bound, use the nearest neighbour result, or improve it by considering all starting vertices. To find a lower bound, delete a vertex, find a minimum spanning tree of the remaining graph, then add the two shortest distinct edges from the deleted vertex.
求上界可使用最近邻结果，或通过考虑所有起点来改进。求下界时，删除一个顶点，求剩余图的最小生成树，再加上从被删顶点出发的两条不同最短边。
State clearly which bound is an upper bound and which is a lower bound. The best upper bound is the smallest one found; the best lower bound is the largest one found.
要清楚说明哪个是上界、哪个是下界。最佳上界是找到的最小值；最佳下界是找到的最大值。
9. Linear Programming | 线性规划
Decision D1 linear programming requires you to define decision variables, write constraints, graph the feasible region, and find the optimal vertex. Use x and y consistently and define them with units in context.
决策数学 D1 线性规划要求定义决策变量、写出约束、画出可行域并找到最优顶点。要始终使用 x 和 y，并结合题意给出单位定义。
Use solid lines for ≤ or ≥ constraints and dashed lines for strict inequalities. Shade out the rejected region or shade in the feasible region, but be consistent with the method stated in the question.
≤ 或 ≥ 约束使用实线，严格不等式使用虚线。可以画出舍弃区域或保留区域，但要与题目要求的方法保持一致。
The objective function can be maximised or minimised. Use the vertex testing method by substituting the coordinates of each feasible vertex, or draw a line with the objective gradient and slide it parallel until it leaves the feasible region.
目标函数可以最大化或最小化。使用顶点检验法，将每个可行顶点的坐标代入，或者沿目标梯度画平行线并移动直到其离开可行域。
For integer solutions, test integer points near the optimal vertex and check that they satisfy all constraints. Do not simply round the coordinates of the optimal vertex, because the rounded point may be infeasible or suboptimal.
对于整数解，检验最优顶点附近的整数点，并确认它们满足所有约束。不要简单地将最优顶点坐标四舍五入，因为取整后的点可能不可行或不是最优。
10. Critical Path Analysis | 关键路径分析
Critical path analysis models a project as activities with
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