Data Structures and Algorithms: cram pack

Formula sheets, the key method steps for every topic and 6 practice problems, with full solutions at the back.

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1Formulas and methods

Graph theory

Handshake lemma
Σ deg(v) = 2|E|
Complete graphs
|E(Kn)| = n(n − 1)/2, |E(Km,n)| = m·n
Regular graph
|E| = n·k/2
Tree and forest
|E| = n − 1, |E| = n − c
Leaves of a tree
L = 2 + Σ (deg(v) − 2) over deg(v) ≥ 3
Euler
circuit: all degrees even; path: exactly 2 odd degrees
Hamilton circuits in Kn
(n − 1)!/2
Dijkstra update
d(v) = min(d(v), d(u) + w(u, v))
Walks of length L
(Mᴸ)ij

Method

  1. Write down the degree of every vertex; the degrees add up to twice the number of edges.
  2. For Euler questions, check that the graph is connected and count the vertices with odd degree.
  3. For trees, use n − 1 edges and the handshake lemma.
  4. For Dijkstra, keep a table of tentative distances and settle the smallest unsettled vertex each round.
  5. For a minimum spanning tree, sort the edges by weight and skip every edge that closes a cycle (Kruskal), or grow one tree from a start vertex (Prim).

2Practice problems

1.Algorithms & complexityMedium

Dynamic programming is useful when a problem has:

  1. AOnly random inputs
  2. BOverlapping subproblems and optimal substructure
  3. CNo subproblems at all, so it can be solved in one step
  4. DA single loop
2.Algorithms & complexityHard

How do n! and 2ⁿ compare for large n?

  1. An! grows faster than 2ⁿ
  2. BThey grow at the same rate
  3. Cn! is O(n²)
  4. D2ⁿ grows faster than n!
3.Data structuresMedium

Which abstract data type does Dijkstra's algorithm use to choose the next vertex to process?

  1. AA FIFO queue
  2. BA set with no ordering
  3. CA stack
  4. DA min-priority queue
4.Data structuresHard

How can you detect whether a singly linked list contains a cycle using O(1) extra memory?

  1. ASort the list and look for duplicates
  2. BMove one pointer by one step and another by two steps and check whether they meet
  3. CStore every visited node in a hash set
  4. DCount the nodes until the count exceeds the list length field
5.Graph theoryMedium

A connected graph has 7 vertices and 9 edges. How many edges k must be removed to leave a spanning tree?

|V| = 7|E| = 9
k =
6.Graph theoryHard
129111519316ABCDE

Prim’s algorithm builds a minimum spanning tree starting from vertex D. Click the edge it adds as edge number 3.

start: Dk = 3
  1. 1edge AB
  2. 2edge AC
  3. 3edge BC
  4. 4edge BD
  5. 5edge CE
  6. 6edge DE
  7. 7edge BE

3Solutions

1.Algorithms & complexity

Answer: B · Overlapping subproblems and optimal substructure

Why this is the answer

Storing results of overlapping subproblems (memoisation or tabulation) avoids repeated work, turning exponential algorithms into polynomial ones.

Answer

B · Overlapping subproblems and optimal substructure
2.Algorithms & complexity

Answer: A · n! grows faster than 2ⁿ

Why this is the answer

For n ≥ 4, n! exceeds 2ⁿ, and the ratio grows without bound because each extra factor of n! is larger than 2.

Answer

A · n! grows faster than 2ⁿ
3.Data structures

Answer: D · A min-priority queue

Why this is the answer

A min-priority queue, usually a binary heap, returns the unvisited vertex with the smallest distance efficiently.

Answer

D · A min-priority queue
4.Data structures

Answer: B · Move one pointer by one step and another by two steps and check whether they meet

Why this is the answer

In Floyd's algorithm, if there is a cycle the fast pointer eventually catches up with the slow one; otherwise the fast pointer reaches null.

Answer

B · Move one pointer by one step and another by two steps and check whether they meet
5.Graph theory

Answer: k = 3

Euler, Hamilton and trees: a connected graph has an Euler circuit (every edge once, back to the start) when every vertex has even degree, and an Euler path that is not a circuit when exactly two vertices have odd degree; the path then runs from one odd vertex to the other. A Hamilton circuit visits every vertex once; Kn has (n − 1)!/2 of them. A tree on n vertices has n − 1 edges, and a connected graph with m edges keeps a spanning tree after removing m − n + 1 edges.

Edges of a spanning tree

7 − 1 = 6

Edges to remove

k = 9 − 6 = 3

Answer

k = 3
6.Graph theory

Answer: edge BC

Shortest paths and spanning trees: Dijkstra’s algorithm settles one vertex per round: the unsettled vertex with the smallest tentative distance. Its distance is then final, and every neighbour v gets d(v) = min(d(v), d(u) + w(u, v)). Kruskal’s algorithm builds a minimum spanning tree by taking the edges from light to heavy and skipping any edge that would close a cycle; Prim’s algorithm grows one tree from a start vertex by always adding the lightest edge that leaves it. A spanning tree of n vertices has n − 1 edges.

Prim’s algorithm

1. DE (3)
2. BD (15)
3. BC (11)
4. AC (9)

edge number 3

BC

Answer

edge BC

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