📚 Further Maths Decision Maths 1 MS: Common Mistakes to Avoid | Further Maths Decision Maths 1 MS 易错点总结
Decision Maths 1 often appears deceptively straightforward; however, beneath simple algorithms lie subtle pitfalls that can cost dear marks. This focused guide unpacks the most frequent errors students make across sorting, networks, linear programming, and critical path analysis, giving you the clarity to spot and sidestep them in the exam.
决策数学1看似简单易懂，但简单的算法背后往往暗藏着容易失分的陷阱。本指南专门剖析学生在排序、网络、线性规划和关键路径分析等模块中最常犯的错误，帮你在考试中清晰识别并巧妙避开这些雷区。
1. Sorting Algorithms: Passes, Comparisons and Swaps | 排序算法：趟数、比较次数与交换的陷阱
Mistake: Stating that bubble sort requires exactly n passes to complete. The algorithm finishes after the first pass with no swaps, but the maximum number of passes is n−1. Students often write ‘n passes’ instead of checking when no swaps occur.
常见错误：认为冒泡排序恰好需要n趟才能完成。实际上，只要某一趟没有发生交换，算法就已结束，最大趟数为n−1。学生常常直接写’n趟’而不验证无交换的提前终止条件。
Pitfall: Confusing comparisons per pass with total comparisons. For bubble sort, pass 1 involves n−1 comparisons, pass 2 involves n−2, giving a total of ½n(n−1) in the worst case. Shuttle sort (insertion sort) can also require up to ½n(n−1) comparisons, but the number of passes equals n−1 and each pass may involve a different number of comparisons and shifts.
易错点：把每趟的比较次数与总比较次数混为一谈。冒泡排序第一趟比较n−1次，第二趟n−2次，最坏情况总比较次数为½n(n−1)。穿梭排序（插入排序）同样最多需要½n(n−1)次比较，但其趟数固定为n−1，每趟的比较和移动次数却有波动。
Watch out: In shuttle sort, forgetting that the ‘sorted’ part grows from the left, and comparing each new item backwards until its correct position is found. A common error is counting a single pass as only one comparison.
注意：在穿梭排序中，常有人忘记已排序部分从左边开始增长，需要将新元素从右向左比较直到找到正确位置。一个典型错误是把整个一趟只计为一次比较。
2. Bin Packing: First-fit and First-fit Decreasing | 装箱问题：首次适应与降序首次适应
Common slip: In first-fit decreasing, failing to sort the items into descending order before applying first-fit. Without the sort, you are performing plain first-fit and will lose marks for the heuristic.
常见疏忽：在使用降序首次适应时，忘记先把所有物品按降序排列就直接装箱。不排序的话就变成了普通首次适应，会丢失该启发式算法的分数。
Critical error: When an item fits into an earlier bin that already has some wasted space, students sometimes open a new bin unnecessarily. First-fit always places the item in the first bin that has enough remaining capacity, even if a later bin could leave more room for future items.
关键错误：当某个物品能放入前面已有剩余空间的箱子时，学生有时会错误地开启新箱子。首次适应规则始终是将物品放入能容纳它的第一个箱子，即使后面某个箱子能留出更多空间也不行。
Lower bound pitfall: The lower bound for the number of bins is ceil(total volume / bin capacity). However, a valid lower bound must also consider individual large items; sometimes a greater lower bound arises from noticing that two large items cannot share a bin. Neglecting this leads to underestimating the true optimum.
下界陷阱：箱子数量的下界是 ceil(总体积 / 箱子容量)，但有效的下界还需考虑单个大件物品的限制；有时两个大件无法放入同一个箱子，这会形成更高的下界。忽略这点会导致低估实际最优解。
3. Minimum Spanning Trees: Prim’s and Kruskal’s Algorithms | 最小生成树：普林姆与克鲁斯卡尔算法
Slip: In Prim’s algorithm, selecting an edge that connects two vertices already in the tree. Always choose the edge of least weight joining a vertex currently in the tree to a vertex not yet in the tree.
失误：在使用普林姆算法时，选择了连接两个已都在树中顶点的边。务必记住：要选择连接树内顶点和树外顶点的权重最小的边。
Mistake with Kruskal: When sorting all edges by weight, picking an edge without checking whether its endpoints are already connected via previously chosen edges, thus accidentally creating a cycle. You must verify that the new edge does not connect two vertices already linked by the partial spanning forest.
克鲁斯卡尔算法错误：按权重排序所有边后，只挑最小边却不检查其两个端点是否已通过之前选择的边连通，无意中形成了环。必须确保新边不会连接已在当前生成森林中连通的两个顶点。
Comparison confusion: Prim’s algorithm builds a single connected tree throughout; Kruskal’s algorithm may maintain a forest of disconnected components that merge later. Recording the order of edge selection incorrectly by mixing the two methods loses accuracy marks.
对比混淆：普林姆算法整个过程中始终保持一棵连通树；克鲁斯卡尔算法则可能先形成多个不连通的子树而后再合并。把两种方法的选边顺序记混，会导致准确性分数丢失。
4. Shortest Path: Dijkstra’s Algorithm | 最短路径：迪杰斯特拉算法
Frequent flaw: Updating working values but forgetting to ‘box’ the smallest temporary label at each stage. Once a vertex receives a permanent label, its value must not be changed; working values may still be lowered.
常见缺陷：更新了工作值，却忘记在每一步将最小的临时标签’固定’（加框）。一旦某个顶点获得永久标签，其值不得再变；而工作值仍可能被降低。
Order slip: Selecting a vertex to make permanent based on a working value that is not globally the smallest among all non-permanent vertices. Always scan all unboxed vertices and choose the one with the current smallest working value.
顺序错误：本应根据全局最小的临时标签来确定永久标签，却选择了一个并非最小的顶点。必须始终扫描所有未加框顶点，选出当前工作值最小的那一个。
Backtracking mistake: When writing the shortest path from source to target, simply listing vertices in reverse order of permanent labelling. The correct method is to trace backwards using the recorded previous vertex for each node, ensuring the path follows actual edges.
回溯错误：在写出从源点到终点的最短路径时，仅仅按永久标签的逆序罗列顶点。正确做法是利用每个节点记录的前驱顶点逐级回溯，确保路径与实际边对应。
5. Route Inspection (Chinese Postman): Pairing Odd Vertices | 中国邮递员问题：奇顶点配对
Critical oversight: Failing to list every possible pairing of the odd vertices when seeking the minimum extra weight. With 4 odd vertices A, B, C, D there are three distinct pairings: (AB+CD), (AC+BD), (AD+BC). Selecting only the two smallest-weight pairs without considering all combinations can give a suboptimal total.
关键疏忽：在寻找最小额外权重时，没有列出所有可能的奇顶点配对组合。对于4个奇顶点A、B、C、D，有三种不同的配对：(AB+CD)、(AC+BD)和(AD+BC)。只挑出两对权重最小的边而忽略全部组合，可能得到次优总距离。
Weight calculation mistake: Adding the direct edge weight between two odd vertices without checking whether a shorter path exists via intermediate vertices. The extra distance must be the shortest path distance between the paired odd vertices, which may be a composite path.
权重计算错误：直接把两个奇顶点之间的边权相加，而没有检查是否存在经过中间顶点的更短路径。额外距离必须是配对奇顶点之间的最短路径长度，可能是组合多段边而成。
Postman tour slip: After adding the repeated edges, simply writing an Eulerian trail without verifying the degrees become all even. The repeated edges might need to be shown explicitly, and the tour should traverse every edge exactly as now laid out.
邮路书写失误：添加重复边后，未验证所有顶点度数变为偶数就直接写欧拉回路。复查时须明确标示重复边，且邮路必须按新布局恰好遍历每条边一次。
6. Critical Path Analysis: Dummy Activities and Floats | 关键路径分析：虚拟活动与浮动时间
Dummy misuse: Inserting dummy activities incorrectly because two activities share the same start and end events but have distinct dependencies. Dummies must only represent logical dependencies that cannot be expressed by activity arrows alone, and each dummy has zero duration.
虚拟活动误用：因为两个活动共享相同起始和结束事件但依赖关系不同而错误插入虚拟活动。虚拟活动只能用来表示无法仅用活动箭头表达的逻辑依赖，且每个虚拟活动持续时间为零。
Float confusion: Mixing total float with independent float. A common mistake is thinking that any activity with non-zero total float can be delayed without affecting any other activity. In reality, total float assumes all preceding activities finish as early as possible and all succeeding start as late as possible; independent float is far more restrictive.
时差混淆：把总浮动时间和独立浮动时间混为一谈。一个常见误区是认为任何总浮动时间不为零的活动都可以推迟而不影响其他活动。实际上，总浮动基于所有前置活动尽早完成、所有后继活动尽晚开始的假设；独立浮动则严格得多。
Critical path identification: Marking a path as critical merely because it ‘looks long’. All activities on the critical path must have zero total float. Students sometimes treat dummy activities as having positive float, but correctly inserted dummies on the critical path also have zero total float.
关键路径识别：仅凭某条路径’看起来很长’就认定其为关键路径。关键路径上的所有活动必须总浮动为零。学生有时以为虚拟活动会有正浮动时间，但实际上正确插入且位于关键路径上的虚拟活动总浮动也为零。
7. Scheduling: Gantt Charts and Worker Allocation | 调度：甘特图与工人分配
Worker doubling: Assigning the same worker to two activities that overlap in time based on early start times. Always draw a Gantt chart with time blocks and check that no worker is scheduled for more than one task at any given instant.
工人重复安排：基于最早开始时间把同一个工人分配到时间重叠的两个活动上。务必绘制甘特图或时间方块图，检查任何时候都不会出现同一工人被安排多项任务的情况。
Resource histogram slip: When drawing a resource histogram from an early-start schedule, presenting a peak requirement as the minimum number of workers needed without attempting to smooth the schedule. Non-critical activities can often be delayed within their total float to reduce the maximum concurrent staffing.
资源直方图错误：在按最早开始时间绘制资源直方图后，直接把峰值需求当作所需最少工人数，而没有尝试对调度进行均衡。通常可以利用非关键活动的总浮动时间将其推迟，从而降低最大并发工人数。
Scheduling algorithm oversight: Using a simple ‘first come, first served’ rule without referring to the ordering implied by the critical path analysis or list scheduling rules. The question may specify a priority list, and ignoring it invalidates the schedule.
调度算法疏漏：使用简单的’先到先服务’规则，而没有参考由关键路径分析或清单调度规则给定的顺序。题目可能明确给出了优先级列表，忽略它会导致整个调度无效。
8. Linear Programming: Feasible Region and Integer Solutions | 线性规划：可行域与整数解
Inequality direction: Reversing an inequality when writing constraints, e.g. using ≤ for a minimum requirement. State clearly what each variable represents and test a sample point to verify the region satisfies the problem statement.
不等式方向错误：在写约束条件时颠倒了不等号，例如对于一个最低要求却使用了≤。应明确每个变量的含义，并用一个样本点检验可行区域是否符合题目描述。
Boundary line slip: Drawing the line x + 2y = 10 as a dotted line for ≤ constraint. Strict inequalities use dashed lines, while ≤ or ≥ use solid lines to indicate the line is included. Shading the wrong side is another frequent slip; always test the origin (0,0) if it is not on the line.
边界线错误：对≤约束却把直线 x + 2y = 10 画成了虚线。严格不等式用虚线，而≤或≥用实线表示边界包含在内。涂错阴影区域也是常见问题；若原点(0,0)不在直线上，务必将它代入检验。
Integer rounding trap: Finding the optimal continuous vertex and merely rounding coordinates to the nearest integers. The true integer solution may lie on a grid point inside the feasible region but not at a rounded vertex. Always test a few integer points near the boundary to confirm the optimum.
整数舍入陷阱：找到最优连续顶点后直接把坐标四舍五入。真正的整数解可能位于可行域内部的某网格点而非舍入后的顶点处。务必在边界附近多测试几个整数点以确定最优解。
9. Matching: Alternating Paths and Maximum Matching | 匹配：交替路径与最大匹配
Starting point mistake: Beginning an alternating path from a matched vertex. An augmenting path must start at an unmatched vertex in one set and end at an unmatched vertex in the other set. Starting from a matched node cannot create the necessary augmentation.
起点错误：从已匹配的顶点开始构建交替路径。增广路径必须从一侧未匹配的顶点出发，终止于另一侧未匹配的顶点。从已匹配节点出发无法达成必要的匹配增加。
Path confusion: Forgetting that edges must alternate strictly between matched and unmatched edges. A common slip is to include two matched edges consecutively, breaking the alternating structure and failing to flip the matchings correctly.
路径混淆：忘记边上必须严格交替匹配边和非匹配边。常见的失误是连续取两条匹配边，破坏了交替结构，导致无法正确翻转匹配状态。
Termination error: Stopping the search as soon as any alternating path is found, without verifying whether it reaches an unmatched vertex at the far end. A valid augmenting path must connect two unmatched vertices; otherwise it does not increase the size of the matching.
终止失误：一旦找到某条交替路径就停止搜索，却没有核实它是否最终抵达一个未匹配的顶点。有效的增广路径必须连接两个未匹配顶点，否则并不能扩大匹配的规模。
10. Algorithm Complexity: Order of Growth | 算法复杂度：增长阶
Comparing incorrectly: Stating that an algorithm with order O(n) will always be faster than one with O(n log n). Big-O notation describes asymptotic behaviour; for small n, constant factors may dominate, but exam questions focus on large inputs. Furthermore, O(n) could describe the best case, while the worst case may be far higher.
比较失当：声称阶为O(n)的算法必定比O(n log n)的算法更快。大O符号描述的是渐近行为；对小规模n，常数因子可能起主导作用，但考题通常关注大输入。另外，O(n)可能只是最好情况，其最坏情况也许要高得多。
Recurrence slip: Mistaking the complexity of binary search as O(n) because each step halves the list. Each halving corresponds to log₂ n steps, so binary search is O(log n). Confusing the base of the logarithm is irrelevant in big-O but forgetting to write ‘log’ entirely costs marks.
递推失误：误以为二分查找的复杂度是O(n)，原因是每一步将列表一分为二。每次分半对应log₂ n步，故二分查找为O(log n)。在大O记号下对数的底数并不重要，但完全忘写’log’会丢分。
Sorting complexities: Assigning O(n log n) to bubble sort in all cases. Bubble and shuttle sorts have worst-case O(n²). QuickSort is O(n²) in the worst case but O(n log n) on average. Merging them up without specifying ‘worst’ or ‘average’ leads to inaccurate statements.
排序复杂度：对所有情况都宣称冒泡排序是O(n log n)。冒泡排序和穿梭排序的最坏情况为O(n²)。快速排序最坏情况是O(n²)，平均情况才是O(n log n)。不加’最坏’或’平均’就混为一谈，会导致表述失准。
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