Online Algorithm

Context: FIT2004_MOC · the classification that decides which algorithms are even admissible for a problem — before complexity is discussed at all · drilled through the streaming $k$-smallest pattern, whose answer is a size-$k$ Heap · contrast the offline Quickselect, which needs every item resident

Quick Revision

• 🎯 Objective: process input one item at a time, committing to a decision without seeing the rest ➔ maintain the answer incrementally instead of computing it once at the end.
• 📦 Core Components: offline ➔ all $N$ items up front, may re-read freely | online ➔ item arrives, act, discard; $N$ may be unknown or unbounded.
• ⚡ Key Constraint: for the $k$-smallest, the heap is a max-heap, not a min-heap ➔ you need the worst admitted item at the root, because the root is the eviction threshold every new arrival is tested against.

📝 How It Works

1. Online vs Offline

• Offline ➔ the whole input is available before the algorithm starts ⟹ it may scan repeatedly, sort, index, or partition destructively — Merge Sort, Quickselect, [[Heap|bottom-up build_heap]].
• Online ➔ items arrive sequentially and the algorithm must hold a valid answer after every arrival, having seen no future item — insertion sort, [[Heap|add/rise]], the size-$k$ heap below.
• Why it matters ➔ (1) the stream may exceed memory ⟹ $\Theta (N)$ space is unaffordable · (2) $N$ may be unknown or infinite ⟹ “sort it first” is not an option · (3) an answer may be required now, at every instant, not after the last item lands.
• The classification precedes the bound ➔ an offline $\Theta (N)$ algorithm is worse than useless on a stream, while an online $\Theta (N\log k)$ one solves it — feasibility outranks asymptotics ➔ Algorithmic Complexity.

2. The Streaming $k$-Smallest

• Structure ➔ a max-heap capped at size $k$, holding the $k$ smallest items seen so far.
• Invariant ➔ after every arrival, the heap contains exactly the $k$ smallest items of the prefix seen so far, and its root is the largest of them.
• Admit ➔ while the heap holds $<k$ items, insert unconditionally in $\Theta (\log k)$.
• Test then act ➔ once full, compare the arrival against the root only: arrival $\ge$ root ⟹ reject in $O(1)$ · arrival $<$ root ⟹ get_max to evict the root, then add the arrival ⟹ $\Theta (\log k)$.
• Why the root alone decides ➔ the root is the largest of the current $k$ smallest, i.e. the weakest member. An arrival $\ge$ root is $\ge$ all $k$ members, so it cannot displace any of them; an arrival $<$ root displaces exactly one, and the root is unambiguously the one that must go. Comparing against the other $k-1$ items is redundant work.
• The mirror ➔ $k$ largest ⟹ min-heap of size $k$, root $=$ smallest admitted $=$ the threshold. Always heap on the opposite extreme to the one you are collecting.
• Answering “the $k$-th smallest” ➔ it is the root at termination; the heap yields the whole $k$-set for free, which Quickselect does not.

3. Cost Profile

• Time $\Theta (N\log k)$ ➔ each of $N$ arrivals costs $O(1)$ to reject or $\Theta (\log k)$ to admit; the reject path dominates in practice once the heap has settled on small values.
• Space $\Theta (k)$, independent of $N$ ➔ this is the property that makes the algorithm online at all; no strategy storing $\Theta (N)$ can process an unbounded stream.
• Single pass ➔ each item is examined exactly once and then discarded — the formal statement of “online”.

⚙️ Core Implementation

🔹 Streaming $k$-smallest with a size-$k$ max-heap

k_smallest_online — one pass, $O(1)$ rejection, no library calls

def k_smallest_online(stream, k):
    heap = MaxHeap()                      # add -> rise, get_max -> sink
    for item in stream:                   # ONE pass; stream length may be unknown
        if heap.length < k:
            heap.add(item)                # fill phase: O(log k)
        elif item < heap.peek_max():      # ONE comparison against the threshold
            heap.get_max()                # evict the current worst
            heap.add(item)                # O(log k) only when it actually wins
        # else: item >= root -> cannot beat any of the k -> DISCARD in O(1)
    return heap                           # the k smallest; root = the k-th smallest

💡 Common Mistake: Reaching for a min-heap because the goal says “smallest” ➔ a min-heap exposes the best item at the root, which tells you nothing about who to evict; you would have to scan all $k$ to find the worst, making every arrival $\Theta (k)$. The heap orders on the opposite extreme to the one being collected.

⚖️ Core Decision Matrix

Strategy	Time	Auxiliary space	Online?	Selection rule
Sort, take first $k$	$\Theta (N\log N)$	$\Theta (N)$	No	the sorted list is wanted anyway
Quickselect	$\Theta (N)$ expected	$O(1)$	No — must partition all $N$	$N$ resident, one rank, mutation allowed
Size-$k$ max-heap	$\Theta (N\log k)$	$\Theta (k)$	Yes	$N$ unknown/unbounded, or $k\ll N$ with a memory cap
Full Heap of $N$, extract $k$	$\Theta (N+k\log N)$	$\Theta (N)$	No	$k$ close to $N$, all items resident
Sorted array, insert each arrival	$\Theta (Nk)$	$\Theta (k)$	Yes	$k$ tiny (2–3), where shifting beats heap overhead

When It Flips: with all $N$ resident and one rank wanted, Quickselect's $\Theta (N)$ beats $\Theta (N\log k)$ — take it. The heap wins the moment either premise fails: unbounded $N$, a memory cap, or a requirement that the answer be valid at every instant. At $k=N$ the heap degenerates to $\Theta (N\log N)$, i.e. Heapsort — so $k\ll N$ is the operating assumption.

📊 Exam Execution Trace & Applied Exercises

Manual Execution Trace

Stream $7,2,9,1,5,8,3$ with $k=3$. Heap contents shown as a set; only the root is ordered.

Step	Arrival	vs root	Action	Heap after	Root (threshold)
0 (Init)	—	—	—	$\{\}$	—
1	$7$	—	fill	$\{7\}$	$7$
2	$2$	—	fill	$\{7,2\}$	$7$
3	$9$	—	fill (now full)	$\{9,7,2\}$	$9$
4	$1$	$1<9$	evict $9$, add $1$	$\{7,2,1\}$	$7$
5	$5$	$5<7$	evict $7$, add $5$	$\{5,2,1\}$	$5$
6	$8$	$8\ge 5$	reject, $O(1)$	$\{5,2,1\}$	$5$
7	$3$	$3<5$	evict $5$, add $3$	$\{3,2,1\}$	$3$

Final: $\{1,2,3\}$ — the $3$ smallest, with the root $3$ being the $3$rd smallest. The threshold is monotonically non-increasing ($7\to 9\to 7\to 5\to 3$ after the fill phase), so rejections get cheaper as the stream runs.

Applied Exercise

Problem: A sensor emits $N=10^9$ readings; report the $k=100$ smallest under a memory cap that forbids storing the stream. Justify the algorithm and quote both bounds.

$$\begin{array}{rl}\text{sort-then-take}& :\Theta (N\log N),\Theta (N)=\Theta (10^9)\text{ space}⟹\text{infeasible}\\ \text{[[Quickselect]]}& :\Theta (N)\text{ time, but requires all }N\text{ resident}⟹\text{infeasible}\\ \text{size-}k\text{ max-heap}& :\Theta (N\log k)=\Theta (10^9\times 7),S_{\text{aux}}=\Theta (k)=\Theta (100)\end{array}$$

Final Extracted Output: the size-$k$ max-heap — the only candidate whose space is $\Theta (k)$ rather than $\Theta (N)$. Note the winner is asymptotically slower in time than Quickselect; the constraint that selects it is space and the online requirement, not speed.

⚠️ Common Mistakes

• 💡 Calling Quickselect online because it is fast ➔ it partitions the entire array, so it needs every item before it can emit anything; $\Theta (N)$ time does not make an algorithm streamable.
• 💡 Comparing the arrival against all $k$ heap items ➔ destroys the $O(1)$ reject path and makes each arrival $\Theta (k)$; the heap exists precisely so that one comparison suffices.
• 💡 Quoting $\Theta (N\log N)$ ➔ the heap is capped at $k$, so operations cost $\Theta (\log k)$; the bound only degenerates to $\Theta (N\log N)$ when $k=\Theta (N)$.

🧠 Active Recall

To collect the $k$ smallest items you use a max-heap. Justify the inversion.

• Hint: Ask which item you need constant-time access to — the best or the worst.

Answer

• Short answer: The operation you repeat is eviction of the worst admitted item, so the worst must be at the root.
• Why: The root is the admission threshold ➔ in a max-heap of the $k$ smallest, the root is their maximum; an arrival $\ge$ root is $\ge$ all $k$ and is rejected in $O(1)$, an arrival $<$ root replaces exactly that root. A min-heap would expose the global smallest — an item you never need to touch — and force a $\Theta (k)$ scan to find the eviction victim.

State the invariant of the size- $k$ heap and show that one comparison per arrival preserves it.

• Hint: Compare the item against the boundary of the admitted set, not its interior.

Answer

• Short answer: The heap holds exactly the $k$ smallest of the prefix seen so far. The root is the boundary, so testing against it is necessary and sufficient.
• Why: Maintenance in two cases ➔ if arrival $\ge$ root then arrival $\ge$ every heap member, so the $k$ smallest of the extended prefix are unchanged ⟹ reject preserves it; if arrival $<$ root then arrival belongs to the new $k$ smallest and the old root does not ⟹ the swap preserves it. No third case exists, so no further comparison can add information ➔ Invariant.

Which of insertion sort, Merge Sort and [[Heap|bottom-up build_heap]] are online, and what does that predict about their use on a stream?

• Hint: Ask whether each holds a valid answer after every single arrival.

Answer

• Short answer: Insertion sort is online; merge sort and bottom-up heap construction are offline.
• Why: Incrementality ➔ insertion sort absorbs a new last element in one pass while keeping the prefix sorted throughout, so it is always answer-valid ➔ Sorting Problem. Merge sort must split the whole array before combining anything, and $\Theta (n)$ build_heap requires all elements placed before sinking begins — on a stream it must be replaced by $n$ online add calls at $\Theta (n\log n)$ ➔ Heap.