The most efficient method to do this would be with a boost converter, preferably one that has current control. Hundreds of LED drivers which are essentially modified power ICs are available nowadays, including ones that operate from single alkaline cells down to 0.9 V e.g. the Micrel MIC2282 (for more look up "single cell led driver") which should be able to drive up to 33 V strings (~8 LEDs), though asking it do that from a single cell may be pushing it
In a previous question, I did actually recommend a power IC, the Linear LT1618 (not outwardly marketed as an LED driver, but an application schematic showed it as one)
Whatever you choose, powering (20) 20 mA LEDs from a single alkaline cell is probably not feasible just because the voltage will dip so low with that load it will undervolt most controllers. If you could use two, you'd probably be in business.
In addition to Russell's answer, I'd like to provide an alternative which I used for my 3kW Strobe light I designed, which uses 24V and does 120A during the pulse, over 48 parallel 12W white LEDs.
In your case you would use a string of LEDs in series as Russell describes, but instead of the LM317 you may use a low-side N channel MOSFET and a "hardware" constant current feedback loop. The feedback consists of a "shunt" resistor and a NPN transistor and another resistor or two.
I found the picture I used as reference for my design, which worked fine. I had my limit at 2.3A per LED though, yours will be much less crazy. Lower current is better for heat issues anyway.
It is very simple, you merely calculate the voltage built up over R3 in the picture so that by Ohms law, your required current will generate 650mV-700mV, enough to turn on the base of the NPN transistor and therefore 'turn off' the MOSFET's gate.
R1 in the picture should be something useful, like 2.2-10K as a pull-up to charge the Gate.
Obviously make sure R3 is rated to handle the current going through it. For 1 Amp and to create 0.7V drop you can use a resistance of 0.7 Ohms, and it will need to be rated 1.5x above the expected dissipation just for good measure, so I^2 * R = ~1.5W-2W rated axial or chip resistor will work fine. I did all of my stuff surface mount, but in my huge strobe light I never went above 5-10% duty cycle in reality, which means I could have down-rated my components significantly, but I kept them as high power packages just in case.
Best Answer
If we take a typical string such as this one made by Ningbo Laihe Christmas Gifts Co.,Ltd, the power supply is 30V (for isolation and so it can use low voltage wire). The total power is only 9W. That's about the power used by just two C6 15V incandescent lamps (series string) that were common when I was a child.
The 1000 lamps could be arranged into series strings of 8 LEDs with a single resistor (perhaps molded into the base or cord) in series with each string. There could be a resistor in series with each lamp of 1/8 the value, but that would be wasteful.
Then put 125 of those series strings in parallel to operate from the 30V source. The 30V would be produced by a switching power supply adapter similar to a phone charger or other modern mains-operated switching supply.