📚 Further Maths Decision Maths 1 AB Question Types Analysis | 进阶数学决策数学1 AB题型解析
Decision Mathematics 1 (D1) is a core module in many A-Level Further Mathematics specifications, particularly Edexcel. It introduces students to a range of algorithms and network problems. Exam questions can often be categorised into two broad types: Algorithm-based questions (Type A) and Graph/Network-based questions (Type B). Understanding the distinction and mastering the techniques for each is key to achieving a high grade. This article breaks down these two major question types, offering detailed analysis, worked examples, and strategic advice to help you tackle D1 with confidence.
决策数学1（D1）是许多A-Level进阶数学考试（尤其是Edexcel考局）的核心模块之一，它向学生介绍了一系列算法和网络问题。考试题目通常可以归为两大类：基于算法的题目（A型）和基于图/网络的题目（B型）。理解这种区分并掌握每种题型的解题技巧是取得高分的关键。本文将详细解析这两类主要题型，提供思路分析、示例讲解和策略建议，帮助你自信地应对D1考试。
1. Overview of AB Classification | AB题型分类概述
Type A questions focus directly on the mechanical execution of algorithms. You will typically be asked to trace sorting algorithms, perform a binary search, apply bin-packing heuristics, or analyse the efficiency of a given procedure. The mark schemes heavily reward accurate step-by-step working and the ability to fill in tracking tables. Type B questions, on the other hand, involve graphs and networks. They require you to find minimum spanning trees using Kruskal's or Prim's, determine shortest paths via Dijkstra's algorithm, solve route inspection problems, or conduct critical path analysis. Many questions also blend algorithmic thinking with graph constructions, such as applying the planarity algorithm or tackling linear programming formulations.
A型题目直接关注算法的机械执行。你通常需要追踪排序算法、执行二分搜索、应用装箱启发式算法或分析给定过程的效率。评分标准非常看重准确的逐步操作以及填写跟踪表的能力。B型题目则涉及图和网络。它们要求你使用Kruskal或Prim算法找到最小生成树、通过Dijkstra算法确定最短路径、解决路径检查问题或进行关键路径分析。许多题目还会将算法思维与图的构造结合起来，例如应用平面性算法或处理线性规划建模。
2. Sorting Algorithms (A-type) | 排序算法（A型）
Sorting algorithms are a staple of D1 exams. The three methods you must know are bubble sort, shuttle sort (insertion sort), and quick sort. For bubble sort, you work through the list comparing adjacent pairs and swapping them if they are out of order. A complete pass scans the entire list; the algorithm terminates when a pass is made without any swaps. Marks are awarded for showing each pass clearly and counting the number of comparisons and swaps. The shuttle sort builds a sorted sublist on the left, picking the next item and shuttling it backward until it reaches the correct position. Quick sort selects a pivot (often the middle value or the first element) and partitions the remaining numbers into those less than the pivot and those greater than the pivot. You must apply the procedure recursively to each sublist.
排序算法是D1考试中的固定内容。你需要掌握的三种方法分别是冒泡排序、穿梭排序（插入排序）和快速排序。冒泡排序中，你遍历整个列表，比较相邻元素并在顺序错误时交换它们。完整的一趟扫描是整个列表；当某一趟没有发生任何交换时算法终止。清楚地展示每一趟并统计比较和交换的次数能够获得分数。穿梭排序在左侧构建已排序的子列表，选取下一个条目并将其向左穿梭，直到到达正确的位置。快速排序则选择一个主元（通常是中间值或第一个元素），然后将剩余的数字划分为小于主元和大于主元的两部分。你必须对每个子列表递归应用该过程。
3. Binary Search & Bin Packing (A-type) | 二分搜索与装箱问题（A型）
Binary search is tested as a precise algorithm that locates a target value in an ordered list. You repeatedly compare the target with the middle term of the current search interval, discarding the half that cannot contain the target. You must maintain a clear record of the left bound, right bound, and pivot index at each step. The algorithm either finds the target or concludes it is absent. For bin packing, you will be asked to pack items of given sizes into bins of fixed capacity using the first-fit, first-fit decreasing, and sometimes full-bin strategies. The first-fit algorithm places each item into the first bin that has enough remaining space. The first-fit decreasing algorithm sorts items into descending order of size before applying first-fit, which usually—but not always—yields a better packing. You must show the contents of each bin and confirm the number of bins used.
二分搜索作为一种精确定位有序列表中目标值的算法会被考查。你需要反复将目标值与当前搜索区间的中项进行比较，并丢弃不可能包含目标值的那一半。你必须在每一步清晰地记录左边界、右边界和主元索引。算法要么找到目标值，要么得出目标值不存在的结论。在装箱问题中，你会被要求将给定大小的物品装入固定容量的箱子中，使用首次适应、首次适应递减，有时还有全箱策略。首次适应算法将每个物品放入第一个有足够剩余空间的箱子中。首次适应递减算法先将物品按大小降序排列，然后再应用首次适应，这通常会——但并不总是——得到更优的装箱结果。你必须展示每个箱子的内容并确认使用的箱子总数。
4. Algorithm Tracing & Efficiency (A-type) | 算法追踪与效率（A型）
Many Algorithm-based questions provide a flowchart or a written algorithm and ask you to trace its operation for a given input. This involves completing a trace table that records the values of variables at each step. Common traps include missing the final loop increment or misreading the condition for loop termination. You could also be asked about the efficiency of algorithms—usually measured by counting comparisons or swaps. For example, you might need to state the maximum number of comparisons for a binary search on a list of size n, which is floor(log₂ n) + 1, or the number of passes in bubble sort (n - 1). Being precise with algorithmic notation and showing all intermediate steps is essential.
许多基于算法的题目会提供一个流程图或一段文字算法，并要求你针对给定输入追踪其运行过程。这涉及完成一个记录每一步变量值的跟踪表。常见的失分点包括遗漏最后一次循环递增，或者误读了循环终止的条件。你还可能被问及算法的效率——通常通过统计比较次数或交换次数来衡量。例如，你可能需要说明对大小为 n 的列表进行二分搜索的最大比较次数是 floor(log₂ n) + 1，或者冒泡排序的趟数为 n - 1。精确掌握算法符号并展示所有中间步骤是至关重要的。
5. Minimum Spanning Trees (B-type) | 最小生成树（B型）
Type B questions often begin with a weighted, undirected graph. To find a minimum spanning tree that connects all vertices with the smallest total weight, you can use Kruskal's algorithm or Prim's algorithm. Kruskal's algorithm sorts all edges by weight and adds them one by one, rejecting any edge that would form a cycle. You must list the edges in the order they are selected and draw the resulting tree. Prim's algorithm grows a tree from an initial vertex, repeatedly adding the cheapest edge that connects a vertex already in the tree to one not yet included. You may be asked to use Prim's on a distance matrix, which requires a different tabular format. Be careful to update the list of nearest distances after each addition and to record the connecting edge.
B型题目常从一个带权无向图开始。为了找到连接所有顶点且总权重最小的最小生成树，你可以使用Kruskal算法或Prim算法。Kruskal算法将所有边按权重排序，然后逐条添加，拒绝任何会形成环的边。你必须按选择顺序列出边，并画出最终的树。Prim算法从一个初始顶点开始生长树，反复添加连接树内顶点与树外顶点的最便宜边。你可能会被要求在一个距离矩阵上使用Prim算法，这需要不同的表格格式。注意在每次添加后更新最近距离列表，并记录连接边。
6. Dijkstra's Shortest Path (B-type) | Dijkstra最短路径（B型）
Dijkstra's algorithm finds the shortest path from a source node to every other node in a weighted graph. You maintain a list of vertices with temporary and permanent labels. Starting at the source, you make its label permanent and update the working values of its neighbours. The process repeats by selecting the vertex with the smallest temporary label, making it permanent, and updating its neighbours' labels. The final answer includes the permanent label for each vertex and the route identified by backtracking. When there are multiple shortest paths, the exam sometimes asks for the number of distinct shortest routes or to trace the path via the edges. Accuracy in arithmetic and label ordering is vital because one small mistake can cascade through the table.
Dijkstra算法用于在带权图中找到从源节点到其他所有节点的最短路径。你维护一个带有临时标签和永久标签的顶点列表。从源点开始，将其标签设为永久，并更新其邻居的工作值。重复此过程：选择具有最小临时标签的顶点，将其设为永久，再更新其邻居的标签。最终答案包括每个顶点的永久标签以及通过回溯确定的路径。当有多条最短路径时，考试有时会要求计算不同最短路径的数量，或者通过边追踪路径。为了保证得分，算术计算和标签顺序的准确性至关重要，因为一个小错误可能会在表格中引发连锁反应。
7. Route Inspection Problem (B-type) | 路径检查问题（B型）
The Chinese postman problem requires you to find the shortest closed walk that traverses every edge at least once. The first step is to identify all vertices with odd degree. If there are no odd vertices, the graph is Eulerian and the optimal route is simply the sum of all edge weights. With exactly two odd vertices, you find the shortest path between them and add its edges as duplicate traversals. When there are more than two odd vertices, you must consider all possible pairings of the odd vertices, find the shortest total duplication weight for each pairing, and choose the minimum. This leads to a semi-Eulerian graph where an Eulerian trail exists. You must explicitly state the route, the total length, and sometimes list the edges that are traversed twice.
中国邮递员问题要求你找到一条至少遍历每条边一次的最短闭合路线。第一步是识别所有奇度数顶点。如果没有奇顶点，该图是欧拉图，最优路线就是所有边权重之和。当恰好有两个奇顶点时，你找到它们之间的最短路径，并将该路径上的边作为重复遍历添加。当奇顶点多于两个时，你必须考虑它们所有可能的配对方式，计算每种配对的最短重复总权重，并选择最小值。这就得到了一个半欧拉图，其中存在欧拉迹。你必须明确说明路线、总长度，有时还要列出被遍历两次的边。
8. Critical Path Analysis (B-type) | 关键路径分析（B型）
Critical path analysis (CPA) models a project as an activity-on-node or activity-on-arc network. You construct a precedence table and then draw the network, ensuring that dependencies are correctly represented. For each activity, you calculate its earliest start time (EST) and earliest finish time (EFT) using a forward pass, and its latest start time (LST) and latest finish time (LFT) via a backward pass. The total float of an activity is LST - EST (or LFT - EFT). Activities with zero total float lie on the critical path(s); any delay in these activities will delay the entire project. The minimum project duration equals the EST value at the sink node. Gantt charts and resource histograms may follow, requiring you to visualise the schedule and consider resource levelling.
关键路径分析（CPA）将项目建模为节点表示活动（activity-on-node）或箭线表示活动（activity-on-arc）的网络。你构建一个前驱关系表，然后绘制网络，确保依存关系得以正确体现。对于每一项活动，你通过顺推计算其最早开始时间（EST）和最早完成时间（EFT），并通过逆推计算其最晚开始时间（LST）和最晚完成时间（LFT）。活动的总浮动时差为 LST - EST（或 LFT - EFT）。总时差为零的活动位于关键路径上；这些活动的任何延误都将延误整个项目。项目的最短工期等于汇节点的EST值。随后可能需要绘制甘特图和资源直方图，这要求你将进度可视化并考虑资源均衡。
9. Linear Programming (AB Hybrid) | 线性规划（AB混合型）
Linear programming (LP) questions straddle both types. The formulation phase involves translating a word problem into algebraic inequalities, defining decision variables, and stating the objective function. This is an algorithmic skill. The graphical solution, however, draws heavily on coordinate geometry and shading the feasible region. You must label constraint lines clearly, use a ruler, and identify the vertices of the feasible region. The optimal solution is found by sliding the objective line or by evaluating vertices. Integer solutions may require extra iteration. While LP is often a standalone topic, its structured method shares the algorithmic precision of Type A and the spatial reasoning of Type B.
线性规划（LP）题目横跨两种题型。建模阶段涉及将文字问题转化为代数不等式、定义决策变量并陈述目标函数。这是一项算法技能。然而，图解法强烈依赖坐标几何和画出可行域阴影的技巧。你必须清晰地标注约束线、使用直尺，并识别可行域的顶点。通过滑动目标函数直线或计算顶点坐标来找到最优解。整数解可能需要额外的迭代。虽然线性规划通常是一个独立主题，但其结构化的方法既具有A型的算法精确性，也具有B型的空间推理能力。
10. Graph Terminology & Notation (B-type) | 图论术语与符号（B型）
y and notation. You need to distinguish between simple graphs, multigraphs, trees, and bipartite graphs. Terms like degree, valency, order, face, planar, Hamiltonian cycle, and Eulerian trail appear frequently. For example, Euler' n matching, an alternating path algorithm is built on these fundamentals. Being able to quickly sketch a subgraph to find a cycle or to test planarity saves time and helps avoid errors in more complex algorithms like Dijkstra or route inspection.
在B型题目中取得成功的关键在于熟练掌握图论术语和符号的使用。你需要区分简单图、多重图、树图以及二分图。度、价、阶、面、平面、哈密顿圈和欧拉迹等术语出现频率很高。例如，连通平面图的欧拉公式 V - E + F = 2 经常被考查。在匹配问题中，交错路径算法就建立在这些基础之上。能够快速绘制一个子图来寻找环或者检验平面性，可以节省时间并避免在Dijkstra或路径检查等更复杂的算法中出现错误。
11. Exam Strategies for AB Types | AB题型的考试策略
When you open the exam paper, initial triage helps. Type A algorithm questions typically require methodical box-filling and are excellent for building momentum and securing marks early. Start with a sorting or binary search question if it appears as the first exercise. Type B network questions, especially those requiring drawing, take more spatial thinking. It is often wise to sketch diagrams in pencil first and trace over them in ink once you are confident. For Dijkstra or Prim, set up a well-organised table before starting. Always read the question preamble carefully: some ask for a table-driven Prim rather than a graphical one. Double-check the number of marks to guide the depth of working you should show—a one-mark efficiency question needs a concise statement, while a six-mark tracing question demands a fully completed trace table.
拿到试卷时，最初的筛选会有所帮助。A型算法题通常需要有条不紊地填表，非常适合建立信心并早早锁定分数。如果试卷的第一题是排序或二分搜索，可以从这里入手。B型网络题，尤其是那些需要作图的题目，需要更多的空间思维。一个明智的做法是先用铅笔画草图，确定无误后再用墨水笔描实。对于Dijkstra或Prim算法，在动笔之前先设置一个条理清晰的表格。务必仔细阅读题目导言：有些题目要求基于表格的Prim而非图形化的Prim。检查题目分值以指导你展示的解题过程深度——一个只有一分的效率问题只需简洁表述，而一个六分的追踪题则要求完整的跟踪表格。
12. Common Pitfalls | 常见错误
A frequent mistake in Type A is omitting the final check step in a binary search or failing to state that the target is not found. In bubble sort, students often miscount the number of passes when no swap occurs after a full run. In Type B, a classic error in Dijkstra's algorithm is updating the label of an already permanent vertex—once a node is permanent, its label must not be changed. For route inspection, forgetting to add the weight of the shortest path between odd vertices results in an underestimate of the total length. In linear programming, failing to test integer points near the optimal continuous solution can lose marks if integer variables are specified. Mind your notation: in critical path analysis, mixing up EST and LFT in a table renders the subsequent float calculations incorrect.
A型题目中常见的失误是在二分搜索中漏掉最后的检查步骤，或者未能声明目标值未找到。在冒泡排序中，学生经常在刚完成一趟完整扫描而未发生交换时，错误地计算了趟数。在B型题目中，Dijkstra算法的一个经典错误是更新一个已经变为永久标签的顶点的标签——一旦一个节点被标记为永久，其标签就不能再改变。对于路径检查问题，忘记加上奇顶点之间最短路径的权重会导致总长度被低估。在线性规划中，如果指定了整数变量，却没有在最优连续解附近测试整数点，就会丢失分数。注意你的符号使用：在关键路径分析中，将表格里的EST和LFT搞混，会使后续的时差计算全部出错。
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