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

In the activity selection problem, which greedy rule always gives a maximum number of non-overlapping activities?

  1. APick the activity that starts earliest
  2. BPick the shortest activity first
  3. CPick the activity that finishes earliest, then repeat with the compatible ones
  4. DPick the activity with the fewest overlaps first, then repeat with the compatible ones
2.Algorithms & complexityHard

The 0/1 knapsack problem solved with dynamic programming takes O(nW) time. Why is this called pseudo-polynomial?

  1. ABecause n is always small
  2. BW is a number, so the time is exponential in the number of bits used to write W
  3. CBecause the algorithm is randomised and its running time varies with each input
  4. DBecause it only works for W = 0
3.Data structuresMedium

How does extract-min work in a binary min-heap?

  1. AMove the last element to the root and sift it down
  2. BRemove the root and shift all elements one place left
  3. CRemove the last element of the array
  4. DSearch all leaves for the new minimum
4.Data structuresHard

Why do databases often use B+ trees rather than plain B-trees for indexes?

  1. AB+ trees use binary internal nodes, which reduces the number of disk reads
  2. BAll records are in linked leaves, which makes range scans efficient
  3. CB+ trees store data only in the root
  4. DB+ trees do not need to be balanced
5.Graph theoryMedium

A connected graph has 14 vertices and 15 edges. How many edges k must be removed to leave a spanning tree?

|V| = 14|E| = 15
k =
6.Graph theoryHard
81381071241214ABCDEFG

Use Dijkstra’s algorithm to find the length d(F) of the shortest path from A to F in the weighted graph.

start: Aend: F
d(F) =

3Solutions

1.Algorithms & complexity

Answer: C · Pick the activity that finishes earliest, then repeat with the compatible ones

Why this is the answer

Choosing the earliest finishing time can be proven optimal by an exchange argument; the other rules have counterexamples.

Answer

C · Pick the activity that finishes earliest, then repeat with the compatible ones
2.Algorithms & complexity

Answer: B · W is a number, so the time is exponential in the number of bits used to write W

Why this is the answer

The running time is polynomial in the value of W but exponential in its length (log W bits); 0/1 knapsack is NP-complete.

Answer

B · W is a number, so the time is exponential in the number of bits used to write W
3.Data structures

Answer: A · Move the last element to the root and sift it down

Why this is the answer

The last element fills the hole at the root and is swapped with its smaller child until the heap property holds, taking O(log n).

Answer

A · Move the last element to the root and sift it down
4.Data structures

Answer: B · All records are in linked leaves, which makes range scans efficient

Why this is the answer

In a B+ tree internal nodes only guide the search and the leaves are linked in key order, so a range query finds the first key and then walks along the leaves.

Answer

B · All records are in linked leaves, which makes range scans efficient
5.Graph theory

Answer: k = 2

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

14 − 1 = 13

Edges to remove

k = 15 − 13 = 2

Answer

k = 2
6.Graph theory

Answer: d(F) = 27

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.

Dijkstra round by round

Each line: the vertex settled, its final distance, and the neighbours whose distance improves.

A (0): B: 8
B (8): C: 21, D: 16, G: 20
D (16): E: 23, F: 28
G (20)
C (21)
E (23): F: 27
F (27)

Shortest path

A → B → D → E → F
d(F) = 27

Answer

d(F) = 27

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